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• University ofCambridge

Mathematics Tripos

Part III

Profinite Groups

Lent, 2020

Lectures byG. R. Wilkes

Notes byQiangru Kuang

mailto:[email protected]

• Contents

Contents

0 Introduction 2

1 Inverse limits 31.1 Categories & Limits . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Inverse limits and profinite groups . . . . . . . . . . . . . . . . . 31.3 Topology on a profinite group/set . . . . . . . . . . . . . . . . . . 41.4 Change of inverse system . . . . . . . . . . . . . . . . . . . . . . 7

1.4.1 Surjective inverse system . . . . . . . . . . . . . . . . . . 71.4.2 Cofinal subsystems . . . . . . . . . . . . . . . . . . . . . . 7

2 Profinite groups 92.1 Zp, the p-adic integers . . . . . . . . . . . . . . . . . . . . . . . . 92.2 The profinite integers Ẑ . . . . . . . . . . . . . . . . . . . . . . . 92.3 Profinite matrix group . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Subgroups, quotients and homomorphisms . . . . . . . . . . . . . 112.5 Generators of profinite groups . . . . . . . . . . . . . . . . . . . . 13

3 Profinite completion 163.1 Residual finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Finite quotients of free groups . . . . . . . . . . . . . . . . . . . . 21

4 Pro-p groups 234.1 Generators of pro-p groups . . . . . . . . . . . . . . . . . . . . . 234.2 Nilpotent groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Invariance of topology . . . . . . . . . . . . . . . . . . . . . . . . 274.4 Hensel’s lemma & p-adic arithmetic . . . . . . . . . . . . . . . . 29

5 Cohomology of groups 345.1 Group rings and chain complexes . . . . . . . . . . . . . . . . . . 345.2 Different projective resolutions . . . . . . . . . . . . . . . . . . . 365.3 Bar resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.4 Cohomology and group extensions . . . . . . . . . . . . . . . . . 405.5 Worked example: Z2 . . . . . . . . . . . . . . . . . . . . . . . . . 435.6 Cohomology of profinite groups . . . . . . . . . . . . . . . . . . . 44

5.6.1 Pro-p groups of cohomological dimension 1 . . . . . . . . 455.6.2 Presentation of pro-p groups . . . . . . . . . . . . . . . . 49

Index 51

1

• 0 Introduction

0 IntroductionQuestion: how can we tell when two objects are different? This is in general adifficult question and to answer it we need a variety of techniques.

• N ∼= Q is easier thatn Q � R.

• To show two finite dimensional vector spaces are not isomorphic we cancompute their dimensions.

• For simplicial complexes, we can compute that homology groups Hn(X).Note that this is a partial invariant, in contrast to dimensiona which is acomplete invariant for finite dimensional spaces.

• We can also consider π1(X). However in contrast to H1(X), π1(X) is notnecessarily abelian. The Adian-Rabin theorem says that there can be noalgorithm which decides if a finitely presented group is trivial or not. Canwe build algorithm which sometimes work? We can solve the problemfor finite groups. For infinite groups we can build upon this idea. Wecan write out the lists of finite quotiesnts of two groups G1 and G2 andcompare them. The question: when does this work?Before we answer this question, note that a list of quotient groups is avery unpleasant object. Instead, we can combine this list into a single“limiting” object, called the profinite completion. This technique worksin other situations:

– p-adic integers Zp being the “limit” of Z/pnZ,– Galois theory: let

K = Q(nth root of unit for all n)KN = Q(nth root of unit for n ≤ N)

Then K =⋃KN and is a Galois extension over Q so we can consider

Gal(K/Q), which is the “limit” of Gal(Kn/Q).– étale fundamental groups in algebraic geometry.

Aside from profinite groups, we will also study group cohomolgy in thiscourse. It is another invariant of groups. This is related to the homology of asimplicial complex and gives abelian invariants. Among other things, it tells ifa group is free. It answers the question: given a group G and an abelian groupA, how many groups E exists such that A E E and E/A ∼= G?

2

• 1 Inverse limits

1 Inverse limits

1.1 Categories & LimitsRecap on categories and limits. Refer to III Category Theory.

1.2 Inverse limits and profinite groups

Definition (profinite completion). Let G be a group. Let N be the categorywhose objects are finite index normal subgroups N Ef G and with an arrowN1 → N2 if and only if N1 ⊆ N2. This is a poset category.

The assignment

N 7→ G/N(N1 → N2) 7→ (G/N1 → G/N2)

is a functor N→ Grp.The limit of this diagram is a group called the profinite completion Ĝ of

G.

Notation. Ĝ is equipped with homomorphisms making the following diagramcommutes:

G/N1 G/N2

In this course we refer to them as projection maps and the horizontal map astransition map. In addition, we have a canonical map i : G → Ĝ which existsby the definition of limit.

We haven’t shown profinite completion exists. We will prove it shortly in amore general context, by showing that it is an exmaple of a particular kind of,in some sense well-behaved, limit.

Definition (inverse system). A poset (J,�) is called an inverse system iffor all i, j ∈ J exists k ∈ J such that k � i and k � j.

Example. In N, N1 ∩N2 is a subgroup of both N1 and N2.

Definition (inverse system, inverse limit). An inverse system (of groups,sets etc) is a functor J→ C where J is the poset category corresponding toan inverse system.

If F : J→ Grp is an inverse system, the limit of F is called the inverselimit of the objects F (j).

Since this is the central subject of this course, we spell out this definitionexplicitly

3

• 1 Inverse limits

Definition. An inverse system of groups, indexed over an inverse system(J,�), consists of

• a group Gj for all j ∈ J ,

• for all i � j, a transition map φij : Gi → Gj such that φii = idGi , φjk ◦φij = φik.

The inverse limit of the system (Gj)j∈J is a group lim←−j∈J Gj with projectionmaps pj : lim←−Gj → Gj such that φij ◦ pi = pj and such that for any Z withqj : Z → Gj , a map q : Z → lim←−Gj such that pj ◦ q = qj .

Definition (profinite group). A profinite group is the inverse limit of aninverse system of finite groups.

Example.

1. The profinite completion of a group G is a profinite group.

2. Zp, the p-adic integers, is lim←−Z/pnZ.

Proposition 1.1. Let (Gj)j∈J be an inverse system of groups indexed byan inverse system J . lim←−Gj exists and is equal to

L = {(gj)j∈J ∈∏j∈J

Gj : ϕij(gi) = gj}.

Proof. Let pj : L → Gj be the restriction of the projection∏Gj → Gj . Then

ϕij ◦ pi = pj . Now let qj : Z → Gj be a cone. There is a unique mapf : Z →

∏Gj such that pj ◦ f = qj and f(Z) ⊆ L.

Note that we do not use any properties of inverse system or posets. Theconstruction works equally well for sets (except that the resulting inverse limitis not a group). We will see that the finiteness and inverse system requirementensures that the construction gives a nonempty set. To do so we need to bringin topology.

1.3 Topology on a profinite group/setGive each finite group Gj in an inverse system the discrete topology. Then give∏Gj the product topology and lim←−Gj ⊆

∏Gj the subspace topology.

∏Gj is

Hausdorff and compact (Tychonoff). It follows that lim←−Gj is Hausdorff. Sincethe conditions defining the subgroup are closed conditions, it is also compact.

Proposition 1.2. If (Xj)j∈J is an inverse system of nonempty finite setsthen lim←−Xj 6= ∅.

Proof. Consider the set

YI = {(xj) ∈∏

XJ : φij(xi) = xj for all i, j ∈ I}

4

• 1 Inverse limits

where I ⊆ J . The YI ’s are closed and⋃I finite YI = lim←−Xj .

To show for I finite, YI 6= ∅, by definition of inverse system exists k ∈ J suchthat k ≤ i for all i ∈ I. Now Xk is nonempty so exists xk ∈ XK . For i ∈ J ,set X = ϕki(xk). For j /∈ I, set xj to be arbitrary. Then this gives a sequence(xi) ∈ YI .

Now use finite intersection property: suppose I1, . . . , Im are are finite, then

YI1 ∩ · · · ∩ YIm ⊇ YI1∪···∪Im 6= ∅

so lim←−Xj =⋃I⊆J finite YI 6= ∅.

It is perhaps psychologically comforting to point out that the topology on aprofinite group is metrisable, thanks to

Proposition 1.3. If (Xi) is a countable family of metric spaces then∏Xi

is a metric space.

Proof. IB Metric and Topological Spaces.

In applications to profinite completions this is often implied by

Proposition 1.4. If G is a finitely generated group then it has only count-ably many finite index normal subgroups.

Proof. Every finite index normal subgroup arises as the kernel of some homo-morphism G → Sn. For a fixed n the homomorphism is determined by theimage of the generators of G, so finitely many.

Proposition 1.5. Let G be a profinite group. Then multiplication µ :G×G→ G and inversion i : G→ G are continuous maps.

Proof. Example sheet 1.

Thus G is a topological group.

Definition. An isomorphism of topological groups is an isomorphism f :G→ H which is also a homeomorphism.

From now on we only consider homomorphism between profinite groupswhich are continuous.

Lemma 1.6. Let H be a topological group and let G = lim←−Gj be an inverselimit of finite groups with projections pj : G→ Gj. Then a homomorphismf : H → G is continuous if and only if pj ◦ f : H → Gj is continuous, ifand only if ker(pj ◦ f) is an open subgroups of H.

Proof. The first iff is by definition of the product topology on∏Gj . For the

second, let fj = pj ◦ f . If fj : H → Gj is continuous then ker fj = f−1j (1) isopen. Conversely, if ker fj is open then f−1j (gj) = h ·ker(fj), where h ∈ f

−1j (gj)

if nonempty, is open for all gj ∈ Gj . Thus f−1j (U) is open for all U ⊆ Gj .

5

• 1 Inverse limits

Proposition 1.7. Let G be a compact topological group. Then a subgroupU ⊆ G is open if and only if it is closed and has finite index.

Proof. Example sheet 1.

Proposition 1.8. Let G = lim←−Gj be an inverse system of finite groups.Then the open subgroups Uj = ker pj form a basis of open neighbourhoods ofthe identity.

Proof. Let V 3 1 be open. By definition of the product topology, V containsa basic open set of the form p−1j1 (Xj1) ∩ · · · ∩ p

−1jm

(Xjm) 3 1, where Xji ⊆Gji open. Then 1 ∈ Xji ⊆ Gji so by shrinking wlog Xji = {1}. Thus 1 ∈ker pj1 ∩ · · · ∩ ker pjm . Now can find k ∈ J such that k ≤ ji for all i. Then1 ∈ Uk ⊆ ker pj1 ∩ · · · ∩ ker pjm ⊆ V .

Corollary 1.9. A basis of open sets in G is {p−1j (gj) : j ∈ J, gj ∈ Gj}.

Corollary 1.10. Let X ⊆ G = lim←−Gj be a subset. Then X is dense in Gif and only if pj(X) = pj(X).

Proof. If X is not dense then exist nonempty open set U such that U ∩X = ∅.wlog U = p−1j (gj). Then gj ∈ pj(G) \ pj(X). Similarly if X is dense and U isnonempty open, wlog U = p−1j (gj), then gj ∈ pj(G) = pj(X) so X ∩U 6= ∅.

Proposition 1.11. Let (Gj) be an inverse system of finite groups andG = lim←−Gj. Let X ⊆ G be a subset. Then X = lim←−Xj where Xj = pj(X).

Proof. Let

X ′ = lim←−Xj = {(gi) ∈∏

Gj : gj ∈ Xj for all j, φij(gi) = gj}

=⋂p−1j (Xj)

=⋂p−1j (pj(X))

which is closed. X ⊆ X ′ so X ⊆ X ′. Let g ∈ G \X. Then exists a basic openset p−1j (gj) ⊆ G \X. Hence X ∩ p

−1j (gj) = ∅, so gj /∈ Xj , so g /∈ X ′.

Corollary 1.12. X is closed if and only if X = lim←−Xj.

Along the same line

Proposition 1.13. Let G be a profinite group. Then

X =⋂

NEoG

XN.

6

• 1 Inverse limits

Proof. Since ker pj form a neighbourhood basis of the identity,

X =⋂p−1j (pj(X)) =

⋂X · ker pj ⊆

⋂NEoG

XN.

Conversely if g /∈ X then can find a neighbourhood p−1j (gj) of g that is disjointfrom X. Then g /∈ X · ker pj .

Example. If Γ is an abstract group, Γ̂ its profinite completion with i : Γ→ Γ̂then pj(i(Γ)) = Γ/Nj = pj(Γ̂), so i(Γ) is dense in Γ̂.

1.4 Change of inverse system1.4.1 Surjective inverse system

Let (Gj)j∈J be an inverse system with transition funcitons ϕij and projectionspj .

Definition (surjective inverse system). An inverse system is surjective ifthe transition maps ϕij are all surjective.

Proposition 1.14. Let (Xj) be a surjective inverse system of nonemptyfinite sets. Then all projections pj : lim←−Xj → Xj are surjective.

Proof. Example sheet 1.

Proposition 1.15. Let (Xj) be an inverse system of finite sets. Then thereexists a surjective inverse system with the same inverse limit.

Proof. Recall that

lim←−Xj ={(xj) ∈

∏Xj : ϕij(xi) = xj

}.

Define Yj = pj(lim←−Xj). Then Yj with transition maps ϕij |Yi form an inversesystem: if yi ∈ Yi then ϕij(yi) ∈ YJ . Then lim←−Yj = lim←−Xj , and this is asurjective inverse system.

1.4.2 Cofinal subsystems

Definition (cofinal). If J is an inverse system, I ⊆ J is cofinal if for allj ∈ J exists i ∈ I such that i ≤ j.

Therefore I is also an inverse system.

Example.

1. In the system of finite index subgroups of Z, one cofinal system is {n!Z}.

2. If k ∈ J , then J≤k = {j ∈ J : j ≤ k} is a principal cofinal system.

3. A cofinal system of Nop is the same as an increasing sequence of integers.

7

• 1 Inverse limits

Definition (linearly ordered inverse system). An inverse system is linearlyordered if it is isomorphic to a subset of Nop.

Proposition 1.16. Let J be a countable inverse system with no initialelement. Then J has a linearly ordered cofinal system.

Proof. Example sheet 1.

Proposition 1.17. Let (Xj) be an inverse system of (finite) sets. Let I ⊆ Jbe a cofinal system. Then lim←−j∈J Xj

∼= lim←−i∈I Xi.

Proof. We prove the proposition for profinite groups. Let G = lim←−j∈J Gj ,H =lim←−i∈I Gi. The map

∏Gj →

∏Gi is a continuous homomorphism and restricts

to a map f : G→ H. Remains to check this is a bijection. Suppose (gj) ∈ ker f ,then pi(g) = gi = 1 for all i ∈ I. For every j ∈ J exists i ∈ I such that i ≤ j sogj = ϕij(gi) = 1. Thus (gj) = 1. For surjectivity, let (gi) ∈ H. For j /∈ I, leti ∈ I be such that i ≤ j and set gj = ϕij(gi). It is well-defined and (gj) ∈ G.

8

• 2 Profinite groups

2 Profinite groups

2.1 Zp, the p-adic integersLet p be a prime. Consider the inverse system

· · · Z/pn · · · Z/p2 Z/p 1

of finite rings. The inverse limit is the profinite ring p-adic integers

Zp = lim←−Z/pn.

An element α ∈ Zp is a sequence (an) of elements of Z/pn such that an =am mod p

n if n ≥ m, where an = α (mod pn) = pn(a).Addition and multiplication are done component-wise. One way to get such

α is to take a ∈ Z and let an be reductions mod pn of a. This gives ι : Z→ Zp.

Definition (pro-p group, pro-p completion). A pro-p group is an inverselimit of p-groups.

The pro-p completion of Γ is

Γ̂(p) = lim←−NEΓ

Γ/N a p-group

Γ/N.

Therefore Zp = Ẑ(p). Usually we suppress ι and regard Z ⊆ Zp.There is a natural metric on Zp: let α = (an), β = (bn). If α = β then

d(α, β) = 0. Otherwise let n be the smallest integer such that an 6= bn and setd(α, β) = p−n. The restriction of this metric to Z is the “p-adic metric” on Z.

The open ball is

B(0, r) = {(an) : am = 0 for p−m ≥ r}= {(an) : am = 0 for m ≤ − logp r}

= ker(Zp → Z/pb− logp rc)

which is an open subgroup of Zp.

Proposition 2.1. Zp is abelian and torsion-free.

Proof. Abelian is obvious. For torsion-free, let α = (an) ∈ Zp with α 6= 0 andmα = 0 for some m > 0. Write m = prs where s is coprime to p. As α 6= 0,exists n such that α 6= 0 (mod pn), i.e. an 6= 0 ∈ Z/pn. As m 6= 0, s 6= 0.Now consider ma (mod pn+r). Claim this is nonzero: if pn+r | man+r thenpn | an+r, hence an+r = 0 (mod pn) = an (mod pn).

Proposition 2.2. The ring Zp is an integral domain.

Proof. Example sheet 1.

2.2 The profinite integers Ẑ

9

• 2 Profinite groups

Proposition 2.3. Ẑ is abelian and torsion-free.

Proposition 2.4. Ẑ has zero divisors.

Theorem 2.5 (Chinese remainder theorem). There is an isomorphism oftopological rings

Ẑ ∼=∏

p primeZp.

Proof. There is a continuous homomorphism Ẑ→ Zp for every p as

Z/pn Z/pm

We thus have a continuous homomorphism f : Ẑ→∏p prime Zp.

f is surjective if and only if im f ⊆∏

Zp is dense, if and only if im f inter-sects all basic open sets of

∏Zp non-trivially. A basic open set has the form

φ−1(x1, . . . , xr) where φ :∏

Zp → Z/pn11 ×· · ·×Z/pnrr . Now invoke the classicalChinese remainder theorem: let m = pn11 · · · pnrr , then we have a commutativediagram

Ẑ∏

Zp

Z/m Z/pn11 × · · · × Z/pnrr

f

φ

∼=

As (x1, . . . , xr) ∈ im(φ ◦ f), have im f ∩ φ−1(x1, . . . , xr) 6= ∅.Now suppose g ∈ Ẑ \ {0}. Then exists m such that the image of g in

Ẑ → Z/m is nonzero. Now use injectivity of the isomorphism to conclude fmust be injective.

2.3 Profinite matrix groupIf R is a commutative ring with 1 then there is a matrix ring Matn(R) of n× nmatrices whose entries are in R. In particular

Matn(Zp) ∼= lim←−Matn(Z/pm)

Matn(Ẑ) ∼= lim←−Matn(Z/m)

for similar argument as above.Define

SLn(R) = {A ∈ Matn(R) : detA = 1}GLn(R) = {A ∈ Matn(R) : detA ∈ R×}

10

• 2 Profinite groups

As det : Matn(Zp)→ Zp is a polynomial so continuous, SLn(Zp) ⊆ Matn(Zp) isa closed subset and is a group under multiplication. We will show in examplesheet that Z×p and Ẑ× are closed subsets of Zp and Ẑ, and in fact they areisomorphic to lim←−(Z/p

m)× and lim←−(Z/m)×. We have

SLn(Zp) = lim←−SLn(Z/pm)

etc. A version of Chinese remainder theorem also holds.Problem: consider the inclusion SLn(Z) ⊆ SLn(Ẑ). How does this inclusion

look like? For example, is this inclusion dense? (the answer is yes, see examplesheet 2). We know from general theory this holds if and only if SLn(Z) →SLn(Z/m) is surjecitve. But this is not obvious at all. For example how can wefind an element that is mapped to ( 7 94 9 ) ∈ SL2(Z/13)?

Another question: do we have SLn(Ẑ) = ŜLn(Z), i.e. does SLn(Z) haveany other finite quotients other than SLn(Z/m)? The answer is no for n = 2(example sheet 2), and yes for n ≥ 3 (hard theorem of Bass-Lazard-Serre).

2.4 Subgroups, quotients and homomorphismsA reminder that we are working in the category of topological groups so sub-groups are closed and homomorphisms are continuous (non-closed subgroup canbe pretty wild: Ẑ ⊇

∏Zp ⊇

∏Z).

Proposition 2.6. A closed subgroup of a profinite group is a profinite group.

Proposition 2.7. Let G = lim←−Gj be a profinite group of a surjective inversesystem, H ≤ G a closed subgroup. Let Hj = pj(H). Then H has finite index(i.e. open) if and only if [Gj : Hj ] is constant for j ∈ I for some cofinalsubsystem I ⊆ J , in which case [G : H] = [Gj : Hj ] for j ∈ I.

Proof. Exercise.

Proposition 2.8. Let G be a profinite group and N Ec G. Then G/Nequipped with the quotient topology is a profinite group.

Proof. Let G = lim←−Gj be a surjective inverse system. Define Nj = pj(N). ThenNj E Gj and define Qj = Gj/Nj . Exists ψij such that the following diagramcommutes:

Gi Gj

Gi/Ni Gj/Nj

φij

ψij

Check that (Qj , ψij) is an inverse system. Let Q = lim←−Qj . There exists acontinuous map

∏Gj →

∏Qj restricting to f : G → Q. (gj) ∈ ker f if

and only if gj ∈ ker(Gj → Qj) = Nj for all j, if and only if g ∈ N . Thus

11

• 2 Profinite groups

ker f = N . By first isomorphism theorem for groups, exists group isomorphismf : G/N → Q making the diagram commute

G Q

G/N

f

f

f is continuous by definition of quotient topology. f is a homeomorphism be-cause G/N is compact and Q is Hausdorff.

Theorem 2.9 (first isomorphism theorem for profinite groups). If G and Qare profinite groups, f : G → Q is a continuous surjective homomorphism,then exists an isomorphism of topological groups f : G/ ker f → Q makingthe following diagram commute

G Q

G/ ker f

f

f

Corollary 2.10. A (closed) quotient of a profinite group is a profinite groupwhen given the quotient topology.

Definition (morphism of inverse system). Let (Gj) and (Hj) be inversesystem of finite groups, indexed over the same poset J . A morphism ofinverse system is a family of group homomorphisms fj : Gj → Hj such thatfor all i ≤ j, the following diagram commutes

Gi Hj

Gj Hj

fi

ϕGij ϕHij

fj

Proposition 2.11. Given a morphism of inverse systems as above, existsa unique continuous homomorphism f : G→ H such that

G H

Gj Hj

f

pGj pHj

fj

commutes for all j.

Proof. Exercise.

12

• 2 Profinite groups

Proposition 2.12. Let (Gj)j∈J , (Hi)i∈I be inverse systems of finite groups,G = lim←−Gj ,H = lim←−Hj. Let f : G → H be a continuous homomorphism.Assume J and I are countable. Then there are cofinal subsystems J ′ ⊆J, I ′ ⊆ I with an isomorphism α : J ′ → I ′ and a morphism fj′ : Gj′ → Hα(j′)of inverse system which induces f .

Proof. We may assume J and I are linearly ordered, i.e. isomorphic to Nop.Also we may assume (Gj)j∈J is surjective. Set J ′ = J . Construct an increasingsequence kn of natural numbers inductively as follows: the composition G →H → Hn is continuous so the kernel is open, hence containing a basic opensubgroup ker pGkn of G. As ker p

Gkn⊆ ker pHn f , exists a quotient map fn : Gkn →

Hn. Increase kn if necessary so kn > kn−1. Set I ′ = {kn}.

2.5 Generators of profinite groups

Definition (topological generating set). Let G be a topological group. Asubset S ⊆ G is a (topological) generating set for G if 〈S〉 is dense in G. G is(topologically) finitely generated (tfg) if it has a finite topological generatingset.

Definition. Let G be a topological group and S ⊆ G. The closed sub-group generated by S is the smallest closed subgroup of G which contains S.Equivalently, it is the intersection of all closed subgroups which contain S.

Exercise. The closed subgroup generated by S is 〈S〉. More generally, theclosure of a subgroup is a subgroup.

Proposition 2.13. If G is a profinite group, U ≤o G then G is tfg if andonly if U is tfg.

Proof. If G = 〈S〉 for some finite S then U ∩ 〈S〉 is finitely generated. As U isopen, U ∩ 〈S〉 is dense in U .

Conversely if U = 〈S〉 and T is a set of coset representatives for U in G,then S ∪ T is a finite generating set for G.

Proposition 2.14. Let G = lim←−Gj be an inverse system of finite groups.Then S ⊆ G is a topological generating set if and only if pj(S) generatespj(G).

Proof. 〈S〉 is dense if and only if pj(G) = pj(〈S〉) = 〈pj(S)〉 for all j.

Proposition 2.15. α ∈ Zp is a generator is and only if α 6= 0 (mod p).

Proof. Let α = (an). α is a generator if and only if an generates Z/pn for all n,if and only if an is coprime to pn, if and only if an is coprime to p.

13

• 2 Profinite groups

If α is a generator then consider multiplication by α, fα : Zp → Zp. Theimage of fα contains 〈α〉 and hence im fα ⊇ 〈α〉 = Zp so exists β such thatαβ = 1. The converse is also true so we can identify the set of generators of Zpwith Z×p .

Z×p is an open subset of Zp and for all n the natural map Z×p → (Z/pn)×is surjective. Thus many elements of Z are invertible in Zp. For example tocompute the inverse of 2 ∈ Z×3 , we can compute its inverse in (Z/3n)× forincreasingly large n to get

2−1 = (. . . , 41, 14, 5, 2) ∈∏

Z/3n

so 2−1 = (. . . , 81, 27, 5, 2) ∈ Z3.

Proposition 2.16. α ∈ Ẑ is a generator if and only if α 6= 0 (mod p) forall p.

By Chinese remainder theorem this set is∏

Z×p , and thus can be identifiedwith Ẑ×. It is a closed subset of Ẑ and just as in the p-adic integer case,Ẑ× → (Z/n)× is surjective for all n.

Theorem 2.17 (Gaschütz’s lemma for finite groups). Let f : G→ H be asurjective homomorphism of finite groups. Assume that G has a generatingset of size d. Then for any generating set {z1, . . . , zd} of H, there exists agenerating set {x1, . . . , xn} of G such that f(xi) = zi.

Proof. We formulate the theorem in terms of generating vectors: (x1, . . . , xd) ∈Gd whose underlying set generates G. f extends to fd : Gd → Hd. Let y be agenerating vector for H and let NG(y) be the cardinality of the set of generatingvectors x for G such that fd(x) = y. We show NG(y) is independent of y, andthen the results follows.

Induction on |G|. Let y be a generating vector for H. Let C be the set of≤ d-generated proper subgroups of G. Then for all x such that fd(x) = y, either〈x〉 = G or 〈x〉 ∈ C. Therefore

| ker fd| = |{x ∈ Gd : fd(x) = y}| = NG(y) +∑C∈C

NC(y).

As ker fd is manifestly independent of y, by induction hypothesis NG(y) isindependent of y.

Theorem 2.18 (Gaschütz’s lemma for profinite groups). Let f : G→ H bea continuous surjective homomorphism of profinite groups. Assume G has atopological generating set of size d. Then for every generating set {z1, . . . , zd}of H, exists a generating set {x1, . . . , xn} of G such that f(xi) = zi.

Proof. wlog G = lim←−j∈J Gj ,H = lim←−j∈J Hj and f is induced by a morphism ofinverse systems fj : Gj → Hj (see lemma below) and the inverse systems aresurjective. Let

Xj = {xj generating vector for Gj : fj(xj) = pHj (z)}

14

• 2 Profinite groups

which is nonempty. (Xj)j∈J forms an inverse system so

lim←−Xj = {x ∈ Gd : x generator of G : f(x) = z}

is nonempty.

Proposition 2.19. Let G be a profinite group and U be a collection ofopen normal subgroups of G which form a neighbourhood basis at 1. ThenG = lim←−U∈U G/U .

Proof. There exists a homomorphism f : G → lim←−U∈U G/U which is surjectivesince G surjects G/U , and injective because U is a neighbourhood basis: forall g ∈ G \ {1} exists V Eo G such that g /∈ V and exists U ∈ U such thatU ⊆ V .

Example. If G is tfg, take U = {Un} where

Un =⋂{normal subgroups of G of index ≤ n}

which is open since the collection is finite.

As a corollary

Lemma 2.20. If G is a tfg profinite group then G = lim←−j∈J Gj where J iscountable.

15

• 3 Profinite completion

3 Profinite completion

3.1 Residual finiteness

Let Γ be an abstract group (usually finitely generated), Γ̂ = lim←−NEfΓ Γ/N its

profinite completion, and a canonical map ι = ιΓ : Γ→ Γ̂.We have seen for Γ = Z this canonical map is an injection. This injection is

sufficiently important that it deserves its own name.

Definition (residually finite). Let Γ be an abstract group. Γ is calledresidually finite if for every γ ∈ Γ \ {1} there exists N Ef Γ such thatγ /∈ N . Equivalently, γ is not in the kernel of Γ→ Γ/N .

Proposition 3.1. Γ is residually finite if and only if ιΓ : Γ→ Γ̂ is injective.

Proposition 3.2. A subgroup of a residually finite group is residually finite.

Proof. Suppose ∆ ≤ Γ. If γ ∈ ∆ \ {1}, exists N Ef Γ such that γ /∈ N . ThenN ∩∆ Ef ∆ and γ /∈ N ∩∆.

A partial converse holds, provided the subgroup has finite index:

Proposition 3.3. Let Γ be an abstract group, ∆ ≤ Γ of finite index. Thenif ∆ is residually finite so is Γ.

Proof. Let γ ∈ Γ \ {1}. If γ /∈ ∆, take

N = CoreΓ(∆) =⋂g∈Γ

g∆g−1 Ef Γ

and γ /∈ N . If γ ∈ ∆ then exists M Ef ∆ such that γ /∈ M . Then M ≤f Γ soN = CoreΓ(M) Ef Γ and γ /∈ N .

Proposition 3.4. Direct product of residually finite groups is residuallyfinite.

Proof. Example sheet 2.

Proposition 3.5. All finitely generated abelian groups are residually finite.

Remark. Finite generation is necessary here. For example Q has no nontrivialfinite quotient.

A source of residually finite groups is matrix group.

Proposition 3.6. SLN (Z) and GLN (Z) are residually finite for all N .

Proof. If A ∈ GLN (Z), take a prime p greater than an entry of A. Then A isnot in the kernel of GLN (Z)→ GLN (Z/p).

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• 3 Profinite completion

Proposition 3.7 (Non-examinable). [Mal’cev’s theorem] Let Γ be a finitelygenerated subgroup of GLN (K) (resp. SLN ,PSLN ) where K is a field. ThenΓ is residually finite.

As a corollary, fundamental groups of surfaces are residually finite (as theyare contained in PSL2(R) by hyperbolic geometry).

Theorem 3.8. Let G and H be tfg profinite groups. Then G ∼= H if andonly if the set of isomorphism types of continuous finite quotients of G andH are equal.

Proof. Let Gn be the intersection of all open normal subgroups of G of index≤ n. Then Gn Eo G and G = lim←−G/Gn. Define Hn similarly. Now G/Gn isa continuous finite quotient of G so it is also a continuous finite quotient of H.Thus exists V Eo H such that G/Gn ∼= H/V . By definition the intersectionof all normal subgroups of G/Gn of index ≤ n is trivial, so upon taking theirpreimages under H → H/V we have Hn ⊆ V . Then

|G/Gn| = |H/V | ≤ |H/Hn|.

By symmetry we have equality so Hn = V and thus G/Gn ∼= H/Hn.To show there exists an isomorphism of inverse systems, let Sn be the set

of isomorphisms G/Gn → H/Hn, which is nonempty. If fn ∈ Sn then it takesnormal subgroups of G/Gn of index ≤ n − 1 to those in H/Hn, so defines anisomorphism Gn−1/Gn → Hn−1/Hn. Thus fn descends to a map φn,n−1(fn) :G/Gn−1 → H/Hn−1 which makes the following diagram commute

G/Gn H/Hn

G/Gn−1 H/Hn−1

fn

φn,n−1(fn)

(Sn, φn,n−1) is an inverse system of nonempty sets so lim←−Sn is nonempty andits element defines an isomorphism of inverse systems.

As a corollary

Theorem 3.9. Let Γ,∆ be finitely generated abstract groups. Then Γ̂ ∼= ∆̂if and only if the set of isomorphism types of finite quotients of Γ and ∆ arethe same.

The following lemma characterises open subgroups of profinite completion

Lemma 3.10. Let Γ be a finitely generated abstract group. Then the opensubgroups of Γ̂ are precisely the subgroups ιΓ(∆) for ∆ ≤f Γ.

Proof. If ∆ ≤f Γ then ιΓ(∆) is closed. It also has finite index so open: supposeΓ =

⋃ri=r gi∆ is a finite (disjoint) union, then

Γ̂ = ιΓ(Γ) =⋃ιΓ(gi∆) =

⋃ιΓ(gi)ιΓ(∆)

Conversely, let U ≤o Γ̂, then U ∩ ιΓ(Γ) is dense in U . Let ∆ = ι−1Γ (U) =ι−1Γ (U ∩ ιΓ(Γ)). Then ∆ ≤f Γ and ιΓ(∆) = U ∩ ιΓ(Γ).

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• 3 Profinite completion

Question: how much can we learn about Γ from Γ̂, if Γ is residually finite?

Proposition 3.11. If Γ̂ ∼= ∆̂, ∆ is abelian and Γ is residually finite then Γis abelian.

Proof. If ∆ is abelian, all its quotients are abelian, so ∆̂ is abelian. ThenιΓ : Γ→ Γ̂ = ∆̂ shows Γ is abelian.

Proposition 3.12. If G and H are finitely generated and Ĝ ∼= Ĥ thenGab ∼= Hab. In particular if G is abelian and H is residually finite thenG ∼= H.

Proof. If G and H have the same finite quotients then they have the same finiteabelian quotients, which are precisely the finite quotients of abelianisation. Thussuffices to show we can recover a finitely generated abelian group G ∼= Zr × Tfrom its finite quotient. Have

r = max{k : G� (Z/n)k for all n}

and T the largest finite abelian group such that G� (Z/n)r × T .

Example (Baumslag). One does not have to go far from abelian groups to showthis fails in general. Let φ : C25 → C25 be the automorphism t 7→ t6 where t is afixed generator. φ has order 5. Form semidirect products G1 = C25 oφ Z, G2 =C25 oφ2 Z. Write Z = 〈s〉 multiplicatively. Claim G1 � G2 but Ĝ1 ∼= Ĝ2.

Proof. Suppose ψ : G2 → G1 is an isomorphism. Then ψ(C25) = C25 soψ(t, 1) = (ta, 1) where ta generates C25. Let ψ(1, s) = (tb, sc). As sc generatesZ, c = ±1. Now compute

ψ((1, s) •2 (t, 1) •2 (1, s−1)) = ψ(φ2(t), 1) = (φ2(ta), 1).

On the other hand

(tb, sc) •1 (ta, 1) •1 (φ−c(t−b), s−c) = (tbφc(ta), sc) •1 (φ−c(t−b), s−c)= (tbφc(ta)φcφ−c(tb), 1)

= (φc(ta), 1)

so φ2(ta) = φc(ta) so 2 = c (mod 5), absurd.To show the profinite completions are isomorphic, note Ĝ1 = C25oφ Ẑ, Ĝ2 =

C25 oφ2 Ẑ: let G1 → Q be a finite quotient. If the composition Z → G1 → Qhas image of order m, then f factors through the finite quotient C25oφ (Z/5m),so the quotients C25 o Z/5m are cofinal in the finite quotients, so

Ĝ1 = lim←−(C25 o Z/5m) = C25 o Ẑ.

Same for G2.

It is worth noting that if we try to compute the same expressions in theprofinite completions, we can merely conclude c is a topological generator ofẐ — but Ẑ has many generators! In fact, for any c ∈ Ẑ× such that c = 2(mod 5) we can define ψ′ : Ĝ2 → Ĝ1, (ta, sb) 7→ (ta, sbc) which is continuous andinjective. As c is a generator, it is also surjective.

18

• 3 Profinite completion

Open problem: If G is finitely generated and residually finite and F is afinitely generated free group, does F̂ ∼= Ĝ imply F ∼= G? Equivalently, doesthere exists a finitely generated residually finite group G and n ∈ N such thata finite group Q is a quotient of G if and only if d(Q) ≤ n, where d is theminimum number of generators?

Proposition 3.13. If F1, F2 are both finitely generated free and F̂1 ∼= F̂2then F1 ∼= F2.

Proof. Immediate.

What about surface groups? They are fundamental groups of genus g ori-entable surfaces and has presentation

Sg = 〈a1, b1, . . . , ag, bg|[a1, b1] · · · [ag, bg]〉.

Can we tell them apart from free groups? If Ŝg = F̂r then Z2g ∼= Zr so r = 2g.What next?

Proposition 3.14 (basic correspondence). Let G1, G2 be finitely generatedresidually finite groups. Suppose Ĝ1 ∼= Ĝ2. Then there is a bijection

ψ : {H ≤f G1} → {H ≤f G2}

such that if K ≤f H ≤f G1 then

• [H : K] = [ψ(H) : ψ(K)].

• K E H if and only if ψ(K) E ψ(H).

• If K E H then H/K ∼= ψ(H)/ψ(K).

• Ĥ ∼= ψ̂(H).

Every finite-sheeted cover of a surface is a surface, so every finite indexsubgroup of Sg is a surface group so has abelianisation Z2g

′ . However Fr has anindex 2 subgroup, which is free of rank 2(r − 1) + 1 by Nielsen-Schreier. Thusfrom the corrspondence we deduce Sg is not free.

The proposition itself follows from the correpondence between subgroups ofG and Ĝ.

Proposition 3.15. Let G be finitely generated residually finite and regardG as a subgroup of Ĝ. Then there is a bijection

{H ≤f G} → {U ≤o Ĝ}H 7→ H

U ∩G← [ U

Furthermore if K ≤f H ≤f G then

• [H : K] = [H : K],

• K E H if and only if K E H,

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• 3 Profinite completion

• If K E H then H/K ∼= H/K,

• H ∼= Ĥ.Proof. Surjectivity is done in Lemma 3.10. To show injectivity enough to showH ∩G = H. Consider the action of G on G/H, inducing a continuous α : Ĝ→Sym(G/H). Then g ∈ StabĜ(H), an open subgroup. If g ∈ G \ H then it isdisjoint from H, so g /∈ H.

Let {gi} be a set of coset representatives of H in G. Then Ĝ =⋃giH. This

is a disjoint union since H ∩G = H, so {gi} is also a coset representative of Hin Ĝ, so there is a natural bijection G/H → Ĝ/H.

If K E H then K = K ∩G E H ∩G = H. H normalises K if and only if Klies in the kernel of the action H → Sym(H/K), so K ⊆ H so K E H.

Any finite quotient K of H is the quotient of H by the open subgroup K.Thus by universal property there is a continuous homomorphism H → Ĥ. It isinjective since for any 1 6= h ∈ H, exists U open in Ĝ such that h /∈ U . Then his not mapped to identity under the composition H → Ĥ → H/H ∩ U .

Definition (Hopf property). A (topological respectively) group G has theHopf property (or is Hopfian, or Hopf) if every (continuous respectively)surjective homomorphism G→ G is an isomorphism.

Usually surjectivity is the easier condition to check — if we know a set ofgenerators then we only have to show that they lie in the image. On the otherhand to show injectivity requires understanding the image of each element.

Proposition 3.16. Let G be a tfg profinite group. Then G has the Hopfproperty.

Proof. Let Gn be the intersection of all open normal subgroups of G of index≤ n. Gn is open and G = lim←−G/Gn. If U ≤o G with [G : U ] ≤ n thenbecause f is surjective, [G : U ] = [G : f−1(U)] ≤ n so Gn ≤ f−1(U). ThusGn ⊆

⋂f−1(U) = f−1(Gn) so f(Gn) ⊆ Gn, inducing a surjective map fn :

G/Gn → G/Gn. Finite groups are Hopfian so fn is an isomorphism. The resultthus follows.

Corollary 3.17. If Γ is finitely generated and residually finite then Γ hasthe Hopf property.

Proof. Let f : Γ→ Γ be a surjection. Then f induces a unique map f̂ : Γ̂→ Γ̂.Γ̂ is tfg and f̂ is surjective because its image is compact and contains a densesubset. The result then follows from the Hopf property of Γ̂.

Proposition 3.18. Let G,H be (topological respectively) groups, G withHopf property. If there exists (continuous respectively) surjections f : G→H, f ′ : H → G then both f and f ′ are isomorphisms.

Proof. f ′ ◦ f : G→ G is an isomorphism by Hopf property so f is injective andan isomorphism. Then f ′ = (f ′ ◦ f) ◦ f−1 is also an isomorphism.

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• 3 Profinite completion

Proposition 3.19. Let Γ be a group and suppose exists a finite quotient Qof Γ such that d(Γ) = d(Q), where d is the minimum number of generators.Then for a free group F , F̂ � Γ̂ unless Γ is free, i.e. Γ ∼= F .

Proof. Let F be a free group with F̂ ∼= Γ̂. Then Q is also a quotient of Fso d(F ) ≥ d(Q) = d(Γ) so exists a surjection f : F → Γ, which induces acontinuous surjection f̂ : F̂ → Γ̂. By Hopf property f̂ is an isomorphism so fis injective.

Example. Non-Hopfian groups are not residually finite. For an example ofnon-Hopfian group, let

BS(n,m) = 〈a, t|ant−1 = am〉

where n,m coprime. Define

f : BS(n,m)→ BS(n,m)y 7→ ta 7→ an

f is a surjection as it surjects the generators. But f is not injective: a does notcommute with tat−1 (which we quote as a fact). But

f([a, tat−1]) = [an, tant−1] = [an, am] = 1

so BS(n,m) is not Hopfian. residually finite.

3.2 Finite quotients of free groups

Theorem 3.20. Free groups are residually finite.

There are two proofs. The first proof in online notes is non-examinable (andomitted). The proof actually shows that

Corollary 3.21. For all p, a free group F is residually p-finite. In particularF injects to its pro-p completion.

We record the second proof here, which is an algorithm to produce finitequotients.

Proof. Let F be free on finite generating set S so F = π1∨S S

1. Write g ∈F \ {1} as a reduced word s1 · · · sn. Write out g “along a line” to get a labelledgraph Y and there is a continuous map Y → X. We seek to make Y a coveringspace of X by adding edges.

For each s ∈ S, the number of vertices having an outgoing s edge is the sameas those having an incoming one. Thus we can add s edges so that every vertexhas exactly one s edge entering and one leaving. This gives a finite covering Yof X so corresponds to a finite index subgroup π1Y of F . g /∈ π1Y since it isnot a loop.

For each s ∈ S, following the edge s is a permutation of vertices to we havean action of F on vertices of Y . g does not fix the initial vertex of Y as g isnot contained in the finite index normal subgroup corresponding to the kernelof the action.

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• 3 Profinite completion

Theorem 3.22 (Marshall Hall’s theorem). Let S ⊆ F be a finite subset. Ify /∈ 〈S〉 then exists a finite group Q and a homomorphism f : F → Q suchthat f(y) /∈ f(〈S〉).

Corollary 3.23. If S does not generate F then exists Q, f such thatf(〈S〉) 6= f(F ).

Remark. Marshall Hall’s theorem actually says that exists H ≤f F such thatS ⊆ H and H = 〈S〉 ∗H ′.

Corollary 3.24. S generates F if and only if S (topologically) generatesF̂ .

Note. As a result 〈S〉 ∩ F = 〈S〉, which generalises the basic correspondence,which holds only for finite index subgroup.

The proof uses monodromy action of the fundamental group. Let X be awedge of circles whose fundamental group is F . If we can find a finite coveringspace Y of X such that S is contained in the image of π1Y then 〈S〉 is containedin the stabiliser of a vertex of the covering action. Similar to the proof ofresidually finiteness of free groups, if y /∈ 〈S〉 then we can construct Y so thatg does not lie in the stabiliser.

To construct the covering space first let Y be the graph with a distinguishedvertex and loops corresponding to words in S going around the vertex. Wewould like to add edges as before so that Y becomes a covering space. Herewe encounter a problem as there might be more than one edge, say a, comingfrom the distinguished vertex. We need to apply Stallings’ fold to identifythe repeated edges, and it is a fact that this procedure does not change thefundamental group.

The formal proof is non-examinable. A worked example can be found in thelecturer’s online notes.

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• 4 Pro-p groups

4 Pro-p groupsRecall that a pro-p group is an inverse limit of finite p-groups — finite groupsof order a power of p. The pro-p completion of Γ is

Γ̂(p) = lim←−NEΓ

Γ/Np-group

Γ/N.

For example Zp = Ẑ(p).

4.1 Generators of pro-p groups

Definition (Frattini subgroup). Let G be a finite group. The Frattinisubgroup of G is

Φ(G) =⋂{M :M maximal proper subgroup of G}.

Proposition 4.1. If f : G → H is a surjective group homomorphism thenf(Φ(G)) ⊆ Φ(H). In particular Φ(G) is a characteristic normal subgroup,i.e. if f : G→ G is an automorphism then f(Φ(G)) = Φ(G).

Proof. Let M be a maximal proper subgroup of H. Then f−1(M) is a propersubgroup and it is maximal: if f−1(M) ⊆ N ⊆ G, assume f−1(M) 6= N . Thenf(N) = H as M is maximal. Therefore G = N · ker f = N as N ⊇ ker f .Therefore Φ(G) ⊆ f−1(M), hence f(Φ(G)) ⊆ M . Taking intersection to getf(Φ(G)) ⊆ Φ(H).

Note that we did not use finiteness anywhere.

Proposition 4.2. Let G be a finite group. For S ⊆ G a subset, TFAE:

1. S generates G.

2. SΦ(G) generates G.

3. SΦ(G)/Φ(G), the image of S in G/Φ(G), generates G/Φ(G).

In other words, the elements of Φ(G) are precisely the non-generators.

Proof. Only 3 =⇒ 1 is nonobvious. Suppose S does not generate G. Then 〈S〉is contained in some maximal proper subgroup. Here we used the crucial factthat G is finite. Since Φ(G) ⊆ M , M/Φ(G) is a proper subgroup of G/Φ(G).Thus SΦ(G)/Φ(G) ⊆M/Φ(G) ( G/Φ(G).

Definition. Let G be a group, H,K subgroups of G and m ∈ Z. Define

HK = {hk : h ∈ H, k ∈ K}

which is a priori a set, but is a subgroup if either H or K is normal, and isa normal subgroup if both H and G are.

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• 4 Pro-p groups

Define the commutator to be

[H,K] = 〈[h, k] = h−1k−1hk : h ∈ H, k ∈ K〉

which is a subgroup and is a normal subgroup if both H and K are normal.Finally define

Hm = 〈hm : h ∈ H〉.

which is a subgroup.

Proposition 4.3. Let G be a p-group. Then

Φ(G) = [G,G]Gp = ker(G→ Gab → Gab/pGab) = 〈[g1, g2]gp3〉.

Note that Gab/pGab is an Fp-vector space, so is isomorphic to (Z/p)d forsome d. Thus G/Φ(G) ∼= Fdp where d = d(G) is the minimum size of a generatingset of G.

Proof. Example sheet 2.

Definition. Let G be a profinite group. Define

Φ(G) =⋂{M :M maximal proper closed subgroups of G},

where M maximal proper closed means that if N is a closed subgroup thenM ⊆ N ⊆ G implies N =M or N = G.

Proposition 4.4. Any proper closed subgroup of a profinite group G iscontained in a proper open subgroup, and hence is contained in a maximalproper closed subgroup, and maximal proper closed subgroups are open.

Proof. Suppose H ≤c G, G 6= H and G = lim←−Gj . Since H is not dense, exists jsuch that pj(H) 6= pj(G). Then p−1j (pj(H)) is open proper and contains H.

Similar to the finite case we have

Lemma 4.5. If f : G → H is a surjective continuous homomorphism ofprofinite groups then f(Φ(G)) ⊆ Φ(H).

Proposition 4.6. If S ⊆ G where G is a profinite group then TFAE

1. S is a tgs for G.

2. SΦ(G) is a tgs for G.

3. SΦ(G)/Φ(G) is a tgs for G/Φ(G).

Proposition 4.7. Let (Gj)j∈J be a surjective inverse system of finite

24

• 4 Pro-p groups

groups. Let G = lim←−Gj. Then

Φ(G) = lim←−Φ(Gj).Proof. Let pj : G → Gj be the projection. Then pj(Φ(G)) ⊆ Φ(Gj) for all j.Hence Φ(G) ⊆ lim←−Φ(Gj).

Now let M be a maximal proper closed subgroup of G. M is open so exists isuch that ker pi ⊆M . Thus ker pj ⊆M for all j ≤ i. Then pj(M) is a maximalproper subgroup of Gj so pj(M) ⊇ Φ(Gj) for all j ≤ i. Thus

M ⊇ lim←−j≤i

Φ(Gj) = lim←−Φ(Gj).

Proposition 4.8. Let G be a tfg pro-p group. Then

Φ(G) = [G,G]Gp

and G/Φ(G) ∼= Fdp where d = d(G).

Later we will see that [G,G]Gp is in fact closed.Note. G/Φ(G) is also denoted H1(G,Fp).Proof. Write G = lim←−Gj as a surjective inverse limit of finite p-groups. ThenΦ(G) = lim←−[Gj , Gj ]G

pj . If g1, g2, g3 ∈ G then

pj([g1, g2]gp3) = [pj(g1), pj(g2)]pj(g3)

p ∈ [Gj , Gj ]Gpj

so pj([G,G]Gp) ⊆ [Gj , Gj ]Gpj for all j so [G,G]Gp ⊆ Φ(G). Now G/[G,G]Gpis tfg abelian and every element has order p. Therefore it is isomorphic toFdp for some d (if a1, . . . , ad tgs of G/[G,G]Gp then {a

n11 · · · a

ndd : n1, . . . , nd ∈

{0, . . . , p − 1}} is a finite dense subgroup). Then since Φ(Fdp) = {0}, we findΦ(G) ⊆ [G,G]Gp as required.

Corollary 4.9. Let f : G → H be a continuous homomorphism of tfgpro-p groups. Then f(Φ(G)) ⊆ Φ(H) and hence there is an induced mapf∗ : G/Φ(G) → H/Φ(H), which is a map of vector spaces over Fp. f issurjective if and only if f∗ is surjective.

Proof. If g1, g2, g3 ∈ G then

f([g1, g2]gp3) = [f(g1), f(g2)]f(g3)

p ∈ Φ(H)

so f(Φ(G)) ⊆ Φ(H).f(G) generates H if and only if f(G)Φ(H)/Φ(H) = f∗(G/Φ(G)) generates

H/Φ(H). As the image of both are compact so closed, the result follows.

Example. Let F = 〈a, b〉, the free group of rank 2 and G = F̂(p). ThenG/Φ(G) = F2p. Let S = {a4b2a, ba−2b}. Map S to G/Φ(G) to test generation:

a4b2a 7→ ( 52 )ba−2b 7→

(−22

)They generate F2p if and only if det

(5 −22 2

)= 14 6= 0, if and only if p 6= 2, 7.

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• 4 Pro-p groups

4.2 Nilpotent groups

Definition. A commutator of length 2 is a commutator

[g1, g2] = g−11 g

−12 g1g2 = g

−11 g

g21 .

Define iteratively a commutator of length n

[g1, g2, . . . , gn] = [g1, [g2, . . . , gn]].

Definition (lower central series). The lower central series of G is the fol-lowing sequence of subgroups:

G1 = G

Gn+1 = [G,Gn] = 〈[g, h] : g ∈ G,h ∈ Gn〉

We sometimes denote Gn by γn(G).

Definition (nilpotent group). A group G is nilpotent of class c if γc+1(G) =1 but γc(G) 6= 1.

For a nilpotent group we have

G = γ1 ≥ γ2 ≥ · · · ≥ γc+1 = 1.

Note. γc(G) is central in G and nilpotent of class 1 is the same as abelian.

Proposition 4.10. γn is fully characteristic in the sense that if f : G→ His a homomorphism then f(γn(G)) ⊆ γn(H).

Proposition 4.11. Subgroups and quotients of nilpotent groups of class care nilpotent of class ≤ c.

Proposition 4.12. A finite p-group is nilpotent.

Exercise. If R is a ring, then the set of upper trianglular n× n matrices with1 on the diagonal is a nilpotent group.

Definition. For G is tfg pro-p group, the lower central p-series is definedby

G1 = G

Gn+1 = [G,Gn]Gpn

γn(G) ⊆ Gn = γ(p)n (G).

Why are we doing this?

26

• 4 Pro-p groups

1. Gn/Gn+1 are vector spaces over Fp.

2. γ(p)n (G) is open in G: inductively Φ(G) = γ(p)2 (G) and Φ(Gn−1) ≤ γ(p)n (G),

and Frattini subgroup of a tfg pro-p group is open.

3. {γ(p)n (G)} forms a neighbourhood basis for the identity: if N Eo G, G/N isa finite p-group so γ(p)n (G/N) vanish for some n. Therefore γ(p)n (G) ⊆ N .

4.3 Invariance of topology

Theorem 4.13. Let G be a tfg pro-p group, H a profinite group and f :G→ H a homomorphism. Then f is continuous.

Corollary 4.14. Let G be a tfg pro-p group. There is no other topology onG making it into a profinite group.

Proof. If τ1 is our given topology and τ2 is another topology such that (G, τ2)is profinite, then the identity homomorphism from τ1 to τ2 is continuous so ahomeomorphism.

“The group structure determines the topology”.Theorem 4.13 follows from

Theorem 4.15. Let G be a tfg pro-p group. Then any finite index subgroupof G is open.

Proof of Theorem 4.13. Suppose f : G → H is a homomorphism. Then for allU Eo H, U has finite index so f−1(H) has finite index and hence open in G.Thus f is continuous.

Lemma 4.16. Let G be a nilpotent group generated by a1, . . . , ad. Thenevery g ∈ [G,G] can be written as

g = [a1, x1] · · · [ad, xd]

for some x1, . . . , xd ∈ G.

Proof. Induction on nilpotency class. c = 1 is trivial. Assume true for nilpo-tency class c− 1. In particular it is true for G/γc(G) so

g = [a1, x1] · · · [ad, xd] · u

for some u ∈ γc(G). We can write u as a product

u =

N∏i=1

[gi, vi]

where gi ∈ G, vi ∈ γc−1(G). Recall the commutator relations

[xy, z] = [x, z]y[y, z]

[x, yz] = [x, z][x, y]z

27

• 4 Pro-p groups

which imply, for v ∈ γc−1(G)

[aiaj , v] = [ai, v][ajv]

[a−1i , v] = [ai, v−1] = [ai, v]

−1

[ai, v][ai, v] = [ai, v2]

[ai, w][ai, v] = [ai, vw]

Now we can rewrite u in the form [a1, v′1] · · · [ad, v′d] so

g = [a1, x1] · · · [ad, xd][a1, v′1] · · · [ad, v′d] = [a1, x1v′1] · · · [ad, xdv′d].

Proposition 4.17. Let G be a tfg pro-p group. Then [G,G] is closed in G.

Proof. Let a1, . . . , ad be a tgs for G. Let

X = {[a1, x1] · · · [ad, xd] : xi ∈ G}.

X is the image of the map

Gd → G(x1, . . . xd) 7→ [a1, x1] · · · [ad, xd]

so is compact so closed. Obviously X ⊆ [G,G]. Let g ∈ [G,G]. Let G = lim←−Gjwhere pj : G → Gj . Then pj(g) ∈ [Gj , Gj ]. Gj is nilpotent as it is a p-group.By the previous lemma

pj(g) = [pj(a1), x1] · · · [pj(ad), xd]

for some x1, . . . , xd ∈ Gj . Hence pj(g) ∈ pj(X) for all j so g ∈ X = X.

Proposition 4.18. Let G be a pro-p group and let K be a finite indexsubgroup. Then [G : K] is a power of p.

Proof. wlog assume K is normal (otherwise replace K by its core). Let [G :K] = m = prm′ where m′ is coprime to p. Consider X = {gm : g ∈ G}. ThenX ⊆ K by Lagrange. X is closed so

X = X =⋂

NEoG

XN.

We will show that gpr ∈ X ⊆ K for all g ∈ G, and this shows [G : K] dividespr (?). Let N Eo G be open normal. Let [G : N ] = ps for some s. Lett = max(r, s). Then gps ∈ N for all g ∈ G. But (m, pt) = pr so exists a, b ∈ Zsuch that am+ bpt = pr. Then

gpr

= (ga)m · (gb)pt

∈ XN.

This holds for all N Eo G so the result follows.

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• 4 Pro-p groups

Proposition 4.19. If G is a tfg pro-p group then [G,G]Gp is closed in G,hence equal to Φ(G).

Proof. Let G{p} = {gp : g ∈ G}, Gp = 〈G{p}〉. As G/[G,G] is abelian,

[G,G]Gp = [G,G]G{p}

and [G,G]G{p} is closed as it is the image of the map

[G,G]×G→ G(x, g) 7→ xgp

Proof of Theorem 4.15. Again wlog look at normal subgroups only. SupposeK Ef G is a counterexample with [G : K] is minimal. Consider

M = [G,G]GpK Ef G

which contains K as a finite index subgroup.Now G/K is a non-trivial p-group so

Φ(G/K) = [G/K,G/K](G/K)p,

which is the image of M in G/K. Hence M is a proper subgroup of G so eitherM = K, so K ≥ [G,G]Gp = Φ(G) open, so K open, or M 6= K, therefore byminimality K is open in M and M is open in G so K is open in G.

4.4 Hensel’s lemma & p-adic arithmeticWe saw earlier that solving the equation ax = 1 in Zp just depends on the imageof a in Z/p. Hensel’s lemma allows us to do so for all polynomials, and gives analgorithm for finding the root.

Lemma 4.20. Let f(x) be a polynomial with Zp coefficients. Then f has aroot in Zp if and only if it has a root modulo Z/pk for all k.

The aim is to reduce just to mod p. To do so we use the method of Hensellifting. As an example let p = 7 and f(x) = x2−2. f(3) = 0 (mod 7) so to finda solution mod 49, consider the element 3 + 7a, 0 ≤ a ≤ 6. Then

(3 + 7a)2 = 9 + 7 · 6a+ 49a2 = 2 + 7(1 + 6a) (mod 49)

so we only have to solve a linear equation since the square term vanishes. a = 1,for example, gives a solution so (3 + 7 · 1)2 = 100 = 2 (mod 49). Next we canconsider 10 + 72 · a ∈ Z/343 etc.

Proposition 4.21 (Hensel’s lemma for square roots). Let p 6= 2 be prime.Suppose λ ∈ Zp is congruent to a nonzero square r21 (mod p). Then existsa unique ρ ∈ Zp such that ρ2 = λ and ρ = r1 (mod p).

Proof. We construct a sequence rk ∈ Z, unique modulo pk, such that

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• 4 Pro-p groups

• rk+1 = rk (mod pk),

• r2k = λ (mod pk).

The first condition can be either interpreted as (zk) forming a Cauchy sequencein Zp, or as (rk) ∈

∏Z/pk compatible with transition functions. In either case

it gives an elemnt ρ ∈ Zp. The second condition then says ρ2 = λ.Suppose we have constructed rk. Consider the elements rk+apk, 0 ≤ a < p.

Since r2k = λ (mod pk), we can write r2k = λ+ bkpk for some bk ∈ Zp. Then

(rk + pka)2 = λ+ pkbk + 2p

kark + p2ka2 = λ+ pk(bk + 2ark) (mod p

k+1)

Now modulo p,bk + 2ark = bk + 2ar1 (mod p)

has a unique root for a (mod p), since 2r1 6= 0 (mod p). Set rk+1 = rk+pka.

Proposition 4.22 (Hensel’s lemma). Let f(x) be a polynomial with Zp coef-ficients and let K ∈ N. Let r ∈ Zp be such that f(r) = 0 (mod pK), f ′(r) 6= 0(mod p). Then exist a unique ρ ∈ Zp such that f(ρ) = 0 and ρ = r(mod pK).

This follows immediately from

Lemma 4.23. For r, a ∈ Zp and k ≥ 1,

f(r + pka) = f(r) + pkaf ′(r) (mod pk+1).

Proof. The statement is linear in f so enough to show for f(x) = xn. Bybinomial formula,

(r + pka)n = rn + npkarn−1 +

n∑i=2

(n

i

)pkiairn−i

= rn + npkarn−1 (mod pk+1)

We can adapt Hensel’s lemma to matrix groups.

Definition. Define filtrations

GL(k)N (Zp) = ker(GLN (Zp)→ GLN (Z/p

k))

= {I + pkA : A ∈ MatN×N (Zp)}

SL(k)N (Zp) = ker(SLN (Zp)→ SLN (Z/p

k))

= {I + pkA : A ∈ MatN×N (Zp),det(I + pkA) = 1}

Proposition 4.24. GL(1)N (Zp) and SL(1)N (Zp) are pro-p groups.

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• 4 Pro-p groups

Proof. WriteGL

(1)N (Zp) = lim←−

k∈ZGL

(1)N (Z/p

k)

and GL(1)N (Z/pk) = {I+pA : A ∈ MatN×N (Z/pk)} has order pN2(k−1). SL(1)N (Zp)

is a closed subgroup of GL(1)N (Zp) and is also a pro-p group.

For the rest of the section we assume p is an odd prime.

Proposition 4.25. The continuous function A 7→ Ap maps GL(k)N (Zp) ontoGL

(k+1)N (Zp). Same for SL.

Slogan: every element in GLk+1n (Zp) has a pth root.

Proof. The proof is by Hensel-like successive approximations. Claim for allr ≥ 1, for all A,

(I + prA)p = I + pr+1A+ pr+2B = I + pr+1A (mod pr+2)

where B is some polynomial of A: for ` ≥ 2 the term(p

p− `

)pr`A`

always has a factor pr+2.Let I+pk+1A ∈ GL(k+1)N (Zp). We show inductively that: for all n ≥ 1, exist

matrices Bn, En both expressible as polynomials in A such that

• Bn+1 = Bn (mod pn),

• (I + pkBn)p = I + pk+1A+ pk+n+1En.

Note that Bn, En’s commute and therefore we can apply binomial theorem. Forn = 1, choose B1 = A. Then

(I + pkA)p = I + pk+1A+ pk+2E1.

Inductively define Bn+1 = Bn − pnEn,

(I + pkBn+1)p = (I + pkBn − pk+nEn)p

= (I + pkBn)p − p · pk+nEn(I + pkBn)p−1 +O(pk+n+2)

= I + pk+1A+ pk+n+1En − pk+n+1En(I +O(pk)) +O(pk+n+2)= I + pk+1A+ pk+n+2En+1

for some En+1. Thus the proposition holds for GL.For SL, suffcies to show detCp = 1 then detC = 1. See example sheet 3

Q10: Z×p = Zp × Cp−1.

Lemma 4.26.

(I + pkA)(I + pkB) = (I + pkB)(I + pkA) = I + pk(A+B) (mod pk+1).

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• 4 Pro-p groups

Proposition 4.27. For all k,

Φ(GL(k)N (Zp)) = GL

(k+1)N (Zp)

andGL

(k)N /GL

(k+1)N

∼= FN2

p .

Proof. By the previous proposition

GL(k+1)N ⊆ (GL

(k)N )

p ⊆ Φ(GL(k)n ).

By the lemma GL(k)N /GL(k+1)N is abelian and of exponent p so is isomorphic

to Fdp for some d. But we have already seen that |GL(k)N /GL

(k+1)N | = pN

2 sod = N2. As Φ(FN2p ) = 1, Φ(GL

(k)N ) ⊆ GL

(k+1)N .

Corollary 4.28. For any k, the continuous map A 7→ Ap induces an iso-morphism

GL(k)N /GL

(k+1)N → GL

(k+1)N /GL

(k+2)N .

Theorem 4.29. If H is a closed subgroup of GL(1)N then d(H) ≤ N2.

Proof. Suffices to show d(H) ≤ N2 for any subgroup H of each finite groupG = GL

(1)N /GL

(k+1)N for each k. For each H ≤ G, for m ≤ k, set

Gm = GL(m)N /GL

(k+1)N

Hm = H ∩Gm

Induction to show d(Hm) ≤ N2: for m = k

Hk ≤ Gk = GL(k)N /GL(k+1)N

∼= FN2

p .

Inductively, let e be the dimension of

Hm/Hm+1 ≤ Gm/Gm+1 ∼= FN2

p .

Have a surjection Hm/Φ(Hm) → Hm/Hm+1 (?). Take e elements h1, . . . , heof Hm which generate Hm/Hm+1. The pth-power map gives an isomorphismGm/Gm+1 ∼= Gm+1/Gm+2 and hence hp1, · · · , hpe are linearly independent inGm+1/Gm+2, thus linearly independent in Hm+1/Φ(Hm+1) (?). By extend-ing to a basis, we can find y1, . . . , yd−e elements of Hm+1 such that Hm+1 =〈hp1, . . . , hpe, y1, . . . , yd−e〉. Then

Hm = 〈h1, . . . , he〉Hm+1 = 〈h1, . . . , he, y1, . . . , yd−e〉

so d(Hm) ≤ d(Hm+1).

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• 4 Pro-p groups

Corollary 4.30 (Non-examinable). There is no closed nonabelian free pro-psubgroup in GLN (Zp).

Sketch proof. F̂(p) has normal subgroups of index pn for all n. These subgroups,by a form of basic correspondence, are free pro-p of rank pn(r−1)+1, absurd.

Compare this with the result that SL2(Z) contains a nonabelian free sub-group.

As a converse we have

Theorem 4.31 (Non-examinable). If G is a pro-p group and exists R suchthat d(H) ≤ R for all H ≤c G then there exists an abelian normal subgroupA ∼= Zep ≤ G (e ≤ R) such that G/A ↪→ GLR(Zp) × F where F is a finitep-group.

33

• 5 Cohomology of groups

5 Cohomology of groups

5.1 Group rings and chain complexesLet G be an abstract group.

Definition (group ring). The group ring ZG is the free abelian group withbasis G, with multiplication given on basis elements by g · h = gh.

ZG is in general noncommutative, but is commutative if G is abelian. Themultiplicative identity is 1e = e. Note that ZG is not necessarily an integraldomain, for example if G has torsion element.

Definition (G-module). A G-module is a ZG-module.

Remark.

1. We only need to think about the action of basis elements.

2. A G-module M is trivial if g ·m = m for all g ∈ G,m ∈M .

Definition. If M1,M2 are G-modules, a morphism of G-modules or a G-linear map is a ZG-module homomorphism M1 →M2.

Definition. Let M1,M2 be G-modules. The Hom-group HomG(M1,M2) isthe set of G-linear maps M1 → G2 with the structure of an abelian group.

Definition (chain complex). A chain complex of G-modules is a sequenceof G-modules and G-module maps

Ms · · · Mn · · · Mt · · · Mtdn dt+1

such that dn ◦ dn+1 = 0 for all n. Sometimes the chain complex is writtenas (Mn, dn).

The complex is exact at Mn if im dn+1 = ker dn. The complex is exactif it is exact at Mn for all t < n < s.

The homology of the complex is the sequence of abelian groupsHS(M•) =ker dS ,Hm(M•) = ker dn/ im dn+1,Ht(M•) =Mt/ im dt+1.

Revision on free/projective modules and free/projective resolution.

Example. Let X be a simplicial complex whose universal cover X̃ is con-tractible. let G = π1X and let Xn be the set of n-simplices of X. Now G actson X̃ with quotient X and no fixed points. Thus the set of n-simplices of X̃ isin bijection with G×Xn. The reduced simplicial chain complex of X̃ takes theform

· · · ZG{X2} ZG{X1} ZG{X0} Z 0

which is a free resolution of Z by G-modules.

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• 5 Cohomology of groups

Definition (group cohomology). Let F• be a projective resolution of Z byG-modules. Let M be a G-module. Apply the functor HomG(−,M) to geta chain complex

HomG(F0,M) HomG(F1,M) · · ·d1 d2

The nth cohomoogy group with coefficients in M is then

Hn(G,M) = ker dn+1/ im dn.

Elements of ker dn+1 and im dn are called n-cocycles and n-coboundariesrespectively.

Example. Let G = Z = 〈t〉 (written multiplicatively). Consider the chaincomplex

0 ZG ZG Z 0d1 ε

where

ε(∑

ngg) =∑

ng

d1(x) = x(t− 1)

It is easy to check this is exact with perhaps the exception ker ε ⊆ im d1. To doso suppose x =

∑Lk=K nkt

k ∈ ker ε so∑nk = 0. Then

x = nLtL−1(t− 1) + nLtL−1 +

L−1∑k=K

nktk

= nLtL−1(t− 1) + (nL + nL−1)tL−2(t− 1) + (nL + nL−1)tL−2 +

L−2∑k=K

nktk

= · · ·= some expression · (t− 1) + (nL + · · ·+ nK)tK−1︸ ︷︷ ︸

=0

Now let M be a G-module and apply HomG(−,M) to get

HomG(ZG,M) HomG(ZG,M) 0d1

Note that HomG(ZG,M) ∼= M as abelian groups so we can rewrite the chaincomplex as

M M 0(t−1)·

so

H0(G,M) = ker d1 = {m ∈M : tm = m} =MG

H1(G,M) =M/ im d1 =M/〈(t− 1)M〉 =MG

which are called invariants and coinvariants of M respectively. Hn(G,M) = 0for n ≥ 2.

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• 5 Cohomology of groups

Proposition 5.1. If G is a free group then Hn(G,M) = 0 for all n ≥ 2.

Non-examinable. Let X be a wedge of circles with π1X = G. The universalcover X̃ is a tree so contractible. The simplicial chain complex of X̃ gives a freeresolution of G-modules of length 1.

Definition (cohomological dimension). A group G has cohomological di-mension n, written cd(G) = n, if Hm(G,M) = 0 for all M for all m > nbut exists M such that Hn(G,M) 6= 0. If no such n exists then cd(G) =∞.

Thus free groups have cohomological dimension 1. The converse is a alsotrue, by a deep theorem of Stallings and Swan.

We now investigate morphisms between complexes of G-modules, whichshould really be done in the context of homological algebra using derived cate-gories. The proofs are omitted.

Definition (chain map). Let (An, αn) and (Bn, βn) be chain complexes ofG-modules. A chain map is a sequence of G-linear maps fn : An → Bn suchthat

fn−1 ◦ αn = βn ◦ fnfor all n.

Proposition 5.2. A chain map (fn) : (An) → (Bn) induces maps f∗ :Hn(A•)→ Hn(B•).

Corollary 5.3. A G-linear map f :M → N induces maps f∗ : H∗(G,M)→H∗(G,N).

Proposition 5.4. If

0 M1 M2 M3 0

is a short exact sequence of G-modules then there is a long exact sequence

· · · Hn(G,M1) Hn(G,M1) Hn(G,M3) Hn+1(G,M1) · · ·

5.2 Different projective resolutions

Theorem 5.5. The definition of Hn(G,M) does not depend on the choiceof projective resolutions.

This follows from the homological algebra fact that projective resolutionsare unique up to quasi-isomorphism. The proof is non-examniable except forthe definition of the chain map between two complexes.

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• 5 Cohomology of groups

5.3 Bar resolutionWe define the so-called bar resolution for the trivial G-module Z. Let G(n) bethe set of symbols [g1|g2| · · · |gn], gi ∈ G. Let Fn = ZG{G(n)}. Define

d0 : F0 → Z∑ngg 7→

∑ng

i.e. the augmentation map. Define

dn : Fn → Fn−1[g1| · · · |gn] 7→ g1[g2| · · · |gn]

+∑

(−1)i[g1| · · · |gigi+1| · · · |gn]

+ (−1)n[g1| · · · |gn−1]

Proposition 5.6. The bar resolution is a free resolution of Z by G-modules.

Non-examinable. Fn is G-isomorphic to Z{Gn+1} (with diagonal G-action) via[g1| · · · |gn] 7→ (1, g1, g1g2, . . . , g1 · · · gn). The latter is a chain complex withboundary maps

(g0, . . . , gn) 7→∑

(−1)i(g0, . . . , ĝi, . . . , gn)

and the complex is acyclic via the obvious chain homotopy

(g0, . . . , gn) 7→ (1, g0, . . . , gn).

Since different projective resolutions give the same cohomology groups wemight reinterpret group cohomology in terms of bar resolution.

Definition. The group of nth cochains of G with coefficients in M is

Cn(G,M) = {Gn →M} = HomG(Fn,M).

The nth coboundary map is

dn : Cn−1(G,M)→ Cn(G,M)φ 7→ ((g1, . . . , gn) 7→ g1 · φ(g2, . . . , gn)

+∑

(−1)iφ(g1g2, . . . , gn)

+ (−1)nφ(g1, . . . gn−1)

The nth cocycle and nth coboundary are

Zn(G,M) = ker dn+1

Bn(G,M) = im dn

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• 5 Cohomology of groups

The nth cohomology is defined to be

Hn(G,M) = Zn(G,M)/Bn(G,M).

We can write out the low-dimensional cocyles and coboundaries explicitly.

H0(G,M) = ker d1 = {m ∈M : gm = m for all g} =MG

is called the set of invariants of M .

ker d2 = {φ : G→M : φ(gh) = gφ(h) + φ(g)}im d1 = {φ : G→M : exists m such that φ(g) = (g − 1)m}

are called crossed homomorphisms and principal crossed homomorphisms.

Example. If M has trivial G-action then

H1(G,M) = Hom(G,M).

Proposition 5.7. Let α : G1 → G2 be a group homomorphism and Ma G2-module. Then there is a natural homomorphism α∗ : Hn(G2,M) →Hn(G1,M).

Proof. Given f ∈ Cn(G2,M), set α∗f ∈ Cn(G1,M) to be the compositionf ◦ αn.

Suppose we have a short sequence of groups

1 H G Q 1.

There is in general no long exact sequence on cohomologies in the style of thesnake lemma.

Example. Let M = Z (with trivial actions) and a short exact sequence 0 →Z→ Z2 → Z→ 0. Then the sequence

H2(Z,Z) H2(Z2,Z) H2(Z,Z)

0 0

cannot be exact as H2(Z2,Z) 6= 0.

There is, however, some exact sequences coming from spectral sequences,namely the five-term exact sequence. In low dimensions they can be describedexplicitly. We state the result below and sketch the proof.

Lemma 5.8. Let H E G and let M be a G-module. Let G act on Cn(H,M)via

(g · φ)(h1, . . . , hn) = gφ(g−1h1g, . . . , g−1hng).

Then this gives an action of G on Hn(H,M). Moreover H acts trivially.

38

• 5 Cohomology of groups

Proof. The first part is an easy computation:

(g · dnφ)(h1, . . . , hn) = gdnφ(g−1h1g, . . . , g−1hng)= g(g−1h1g)φ(g

−1h2g, . . . , g−1hng)

+∑

(−1)igφ(g−1h1g, . . . , g−1hihi+1g, . . . , g−1hng)

+ (−1)ngφ(g−1h1g, . . . , g−1hn−1g)= h1(g · φ)(h2, . . . , hn)

+∑

(−1)id(g · φ)(h1, . . . , hihi+1, . . . , hn)

+ (−1)nd(g · φ)(h1, . . . , hn−1)= (dn(g · φ))(h1, . . . , hn)

To show H acts trivially, we show if h ∈ H, φ ∈ Zn(G,M) then h · φ− φ ∈im dn. Induction on n. n = 1,

0 = (d2φ)(h1, h2) = h1φ(h2)− φ(h1h2) + φ(h1)

soφ(h1h2) = h1φ(h2) + φ(h1)

then

(h · φ− φ)(h1) = (h · φ)(h1)− φ(h1)= hφ(h−1h1h)− φ(h1)= h(h−1φ(h1h) + φ(h

−1))− φ(h1)= h1φ(h) + φ(h1) + hφ(h

−1)− φ(h1)= h1φ(h)− φ(h)= (h1 − 1)φ(h) ∈ im d1

The induction process is another messy calculation and is left as an exercise.

Note if G has trivial action and φ ∈ C1(H,M) then

(g · φ)(h) = φ(g−1hg)

so φ ∈ H1(H,M)G if and only if φ(h) = φ(g−1hg) for all g ∈ G,h ∈ H. Such aφ : H →M is called a G-invariant homomorphism.

Theorem 5.9 (inflation-restriction exact sequence). Let H E G and Q =G/H. Let M be a G-module. Then there is an exact sequence

0 H1(Q,MH) H1(G,M) H1(H,M)G

H2(Q,MH) H2(G,M)

Note that H1(H,M)G = H1(H,M)Q by the previous lemma.

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• 5 Cohomology of groups

Non-examinable. We only define the maps appearing in the sequence. This isdone via the restriction map

Hn(G,M)→ Hn(H,M)G

φ 7→ φ|Hn

the inflation map

Hn(Q,MH)→ Hn(G,M)φ 7→ φ ◦ qn

where q : G → Q, and the transgression map Tg : H1(H,M) → H2(Q,MH)defined as follow: choose a set-theoretic section s : Q → G, i.e. a transversal,with s(1) = 1. Define

ρ : G→ Hg 7→ g · s(gH)−1

If φ : H →M is a Q-invariant cocycle, define

Tg(φ) : G2 →M(g1, g2) 7→ φ(ρ(g1)ρ(g2))− φ(ρ(g1g2))

Tg(φ) descends to Q2 →M .

Corollary 5.10 (Hopf formula). Let F be free, R E F and Q = F/R. If Ais abelian with trivial F -action then

H2(Q,A) ∼={F -invariant homomorphisms R→ A}{restrictions of homomorphisms F → A}

.

Example. SupposeQ = 〈x1, . . . , xd|r1, . . . , rn〉 is a presentation, so F = F{x1, . . . , xd}, R =〈〈r1, . . . , rm〉〉. Then

d(H1(Q,Z)) ≤ dd(H2(Q,Z)) ≤ m

5.4 Cohomology and group extensionsLet E be a group, with an abelian normal subgroup M . Let G = E/M . Suchan E is called an extension of G by M . Two extensions are equivalent if thereis a commutative diagram of homomorphisms

1 M E G 1

1 M E′ G 1

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• 5 Cohomology of groups

Lemma 5.11. Equivalent extensions are isomorphic as groups.

Proof. Same as five lemma.

Note that M comes with the structure of a G-module: the conjugation actionof E on M descends to a G-action. If this action is trivial, the extension is calleda central extension.

Given a G-module M , the group extension problem is concerned about theclassification of the extensions. We can always form the semidirect productE =M oG. It is sometimes also called a split extension.

Definition (splitting). The splitting of an extension E is a group homo-morphism G→ E that is a section to E → G.

Proposition 5.12. Extensions which have a splitting are equivalent to MoG.

Proof. Set M oG→ E, (m, g) 7→ i(m)s(g) where s is a section of E → G.

Let E be an arbitrary extension of G by M . Let s : G→ E be a set-theoreticsection with s(1) = 1. To measure how far s is from being a homomorphism,consider the function

φ(g1, g2) = s(g1)s(g2)s(g1g2)−1.

Then s is a homomorphism if and only if φ is constant. The image of φ is in Mso φ : G2 →M is an element of C2(G,M). Claim that in fact φ ∈ Z2(G,M).

Proof. We compute s(g1)s(g2)s(g3) in two ways:

s(g1)s(g2)s(g3) = φ(g1, g2)s(g1g2)s(g3)

= φ(g1, g2)φ(g1g2, g3)s(g1g2g3)

s(g1)s(g2)s(g3) = s(g1)φ(g2, g3)s(g2g3)

= s(g1)φ(g2, g3)s(g1)−1s(g1)s(g2g3)

= s(g1)φ(g2, g3)s(g1)−1φ(g1, g2g3)s(g1g2g3)

soφ(g1, g2)φ(g1g2, g3) = s(g1)φ(g2, g3)s(g1)

−1φ(g1, g2g3).

Recognising the first three terms on RHS as the action of G on M and convertto additive notation in M , we get

φ(g1, g2) + φ(g1g2, g3) = g1φ(g2, g3) + φ(g1, g2, g3).

In addition φ is a normalised cocycle, i.e.

φ(1, g) = φ(g, 1) = 0.

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• 5 Cohomology of groups

If we had chosen a different section s′ : G→ E, consider ψ(g) = s′(g)s(g)−1so ψ ∈ C1(G,M). Have

s′(g1)s′(g2) = ψ(g1)s(g1)ψ(g2)s(g2)

= ψ(g1)s(g1)ψ(g2)s(g1)−1s(g1)s(g2)

= ψ(g1)s(g1)ψ(g2)s(g1)−1φ(g1, g2)s(g1g2)

= ψ(g1)s(g1)ψ(g2)s(g1)−1φ(g1, g2)

and a comparison with s′(g1)s′(g2) = φ′(g1, g2)s′(g1g2) shows [φ] ∈ H2(G,M)is well-defined. This shows part of

Theorem 5.13. Let G be a group and M a G-module. There exists abijection

{equivalence classes of extensions of G by M} ←→ H2(G,M).

Proof. Let to construct the inverse map. Let [φ] ∈ H2(G,M) where φ is anormalised cocycle. Define a group structure on the set M ×G by

(m1, g1) · (m2, g2) = (m1 + g1 ·m2 + φ(g1, g2), g1g2).

By the assumption on φ, this defines a group structure with identity (0, 1) andthe inverse of (m, g) is (−g−1 · (m+ φ(g, g−1)), g−1). This is an extension.

If we choose a different normalised cocycle φ′ such that φ−φ′ = d2ψ, then theobvious map Eφ → Eφ′ , (m, g) 7→ (m+ ψ(g), g) is an equivalence of extensions.

It is an exercise to check the two maps are inverses to each other.

One result that is used in the proof is

Lemma 5.14. Every cohomology class is represented by some normalisedcocycle.

Proof. Let φ ∈ Z2(G,M). Let ψ(g) = φ(1, g). Then φ− d2ψ is normalised:

(φ− d2ψ)(1, g) = φ(1, g)− (φ(1, g)− φ(1, g) + φ(1, 1)) = φ(1, g)− φ(1, 1)(φ− d2ψ)(g, 1) = φ(g, 1)− gφ(1, 1)

and they both vanish since φ is a cocycle.

We may recover Hopf formula from this identification. Suppose G has pre-sentation 〈x1, . . . , xn|r1, . . . , rm〉. Then take F to be the free groups on xi’s andR = ker(F → G). Suppose E is a central extension of G by A. Choose somepreimages xi ∈ E of generators xi of G. Let ri be the element of E given byreplacing each occurrence of xi in ri with xi. Then ri ∈ A = ker(E → G), sayai. Write down a group presentation

E = 〈x1, . . . , xn, A|r1 = a1, . . . , rm = am, A central, relations of A〉

There exists a natural diagram of exact rows

A E G 1

1 A E G 1

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• 5 Cohomology of groups

It follows that A ↪→ E and E ∼= E. Can try to define F -invariant homomorphismR → A by ri 7→ ai. Fact (non-examinable): this is a well-defined F -invarianthomomorphism if and only if A → E is an injection. We made a choice ofpreimages xi of xi in E. A different choice x′i differ from xi by an elementbi ∈ A. Then xi 7→ bi gives a homomoprhism f : F → A and r′i = rif(bi).

Example. Let G = 〈x1, x2|x1x2x−11 x−12 x1〉. We show H2(G,Z) = 0. Any

central extension of G by Z has a presentation

E = 〈x1, x2, t|x1x2x−11 x−12 x1 = t

k, t central〉.

Now we can make a substitution x1 7→ x1t−k = x′1,

E ∼= 〈x′1, x2, t|x′1tk · x2(x′1tk)−1x−12 (x′1tk) = tk, t central〉 = E′

E and E′ are equivalent as extensions and we can simplify E′

〈x′1, x2, t|x′1x2(x′1)−1x−12 x′1 = 1, t central〉 = Z×G

It follows that H2(G,Z) = 0.

5.5 Worked example: Z2

Let T = Z2 = 〈a, b〉. We will classify all central extensions of T by Z. Startwith a free resolution derived from topology

0 ZT (ZT )2 ZT Z 0β α ε

where

• ε is the augmentation map.

• α(x, y) = x(a− 1) + y(b− 1).

• β(z) = (z(1− b), z(a− 1)).

Exactness can either be derived from topology (square tiling of the plane), ordirect computation.

Apply HomZT (−,Z) to get

HomZT (ZT,Z) HomZT ((ZT )2,Z) HomZT (ZT,Z) 0α∗ β∗

where

β∗(f)(z) = f(z(1− b), z(a− 1))= f(z − zb, 0) + f(0, za− z)= (b− 1) · f(z, 0) + (1− 1) · f(0, z)= 0

so β∗ = 0. Similarly α∗ = 0. Thus

Hi(T,Z) =

Z i = 0Z2 i = 1Z i = 2

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• 5 Cohomology of groups

To get extensions, we turn to bar resolutions. We seek a chain map

. . . ZT{T 2} ZT{T} ZT Z 0

0 ZT (ZT )2 ZT Z 0

d2

f2

d1

f1

ε

f0

β α ε

Obviously f0 = id. f1 should satisfy αf1 = f0d1 = d1. Want to find an element(xr,s, yr,s) ∈ (ZT )2 such that

α(xr,s, yr,s) = d1([arbs]) = arbs − 1 = (ar − 1)bs + (bs − 1)

then define f1 by [arbs] 7→ (xr,s, yr,s). By commutativity of T ,

xr,s =ar − 1a− 1

bs = S(a, r)bs

yr,s =bs − 1b− 1

= S(b, s)

where

S(a, r) =

{1 + a+ · · ·+ ar−1 r ≥ 0−(a−1 + · · ·+ ar) r < 0

So f1([arbs]) = (S(a, r)bs, S(b, s)). By a messy calculation we similarly find

f2([arbs|atbu]) = S(a, r)bsS(b, u).

A cohomology class p ∈ Z ∼= HomZT (ZT,Z) is represented by the 2-cochaingiven by the composition

(arbs, atbu) 7→ S(a, r)bsS(b, u) 7→ pru

so the group structure on the set Z× T corresponding to this cochain is

(m, arbs) · (n, atbu) = (m+ n+ pru, ar+tbn+s).

More concretely, this group has a 3-dimensional representation

(m, arbs) 7→

1 pr m0 1 s0 0 1

The group of central extensions of T by Z is generated by p = 1.

5.6 Cohomology of profinite groupsThe cohomology theory works for profinite groups.

Definition (G-module). Let G be a profinite group. A finite G-module isa finite abelian group M with a continuous G-action G×M →M .

To avoid defining the group ring of a profinite ring, we use the ad hocdefinition

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• 5 Cohomology of groups

Definition. Let G be a profinite group and M a finite G-module. Define

Cn(G,M) = {Gn →M continuous}

and dn given by the same formula as before. DefineHn(G,M) = ker dn+1/ im dn.

Course convention: all general results in this chapter and example sheet 4may be assumed to hold for profinite groups, where all groups are profinite, allfunctions are continuous and all modules are finite.

Example. An extension ofG byM is a profinite group E withM E E such thatE/M ∼= G. Equivalence of extensions is a continuous homomorphism respectingM and G. Then there exists a bijection between equivalent classes of extensionsand H2(G,M).

One can treat profinite groups as discrete groups, but the resulting cohomol-ogy theory is horrid. Another question: why finite modules only? For exampleconsider the exact sequence of Ẑ-modules with trivial Ẑ-action

0 Z Q Q/Z 0

HaveH1(Ẑ,Z) = H1(Ẑ,Q) = 0

as the continuous map from Ẑ → Z has compact image. H1(Ẑ,Q/Z) ∼= Q/Z.But we should have a long exact sequence, which implies that H2(Ẑ,Z). Thusfree profinite groups Ẑ does not have “cohomologicial dimension” 1.

5.6.1 Pro-p groups of cohomological dimension 1

For simplicity in this section assuming all pro-p groups are tfg. The aim of thissection is to prove Stallings-Swan for tfg pro-p groups, i.e. Proposition 5.1 andits converse:

Theorem 5.15. A tfg pro-p group G is free if and only if cd(G) = 1, if andonly if H2(G,Fp) = 0.

Note that for any non-trivial tfg pro-p group G, in particular free pro-pgroups, cd(G) ≥ 1 as

H1(G,Fp) ∼= Hom(G,Fp) 6= 0.

We first show the equivalence of the characterisations

Theorem 5.16. Let G be a pro-p group. Then

cd(G) = max(n : Hn(G,Fp) 6= 0).

Definition (simple). A G-module M is simple if the only G-submodulesare 0 and M .

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• 5 Cohomology of groups

Proposition 5.17. Fix n ≥ 0. Let G be a profinite group and supposeHn(G,S) = 0 for all simple finite G-modules S. Then Hn(G,M) = 0 forall finite M .

Proof. Suppose for contradictionM of minimal size has nonvanishingHn(G,M).M is not simple so exists G-submodule N ≤ M , giving rise to a short exactsequence

0 N M M/N 0

of G-modules. By using long exact sequence and minimality of M , Hn(G,M) =0, absurd.

Definition (p-primary component). Let M be a finite G-module. Let Mpbe the Sylow p-subgroup of M , called the p-primary component of M . ThenM =

⊕pMp.

Proposition 5.18. Let G be a pro-p group, M a finite G-module. Then

Hn(G,M) = Hn(G,Mp)

for n ≥ 1.

Proof. Write M = Mp ⊕ M ′ where M ′ is the direct sum of other q-primarycomponents. Then

Hn(G,M) = Hn(G,Mp)⊕Hn(G,M ′)

(finite p-group case is an exercise in example sheet 4) and we show Hn(G,M ′) =0. Let φ : Gn → M ′ be a continuous function. Claim φ factors as Gn →(G/K)n →M ′ for some K Eo G.

Proof. We want to find K such that each fibre of φ is a union of cosets ofKn. For each m ∈ M ′, φ−1(m) is open and closed, so can be written asφ−1(m) =

⋃(ai + K

ni ) where Ki Eo G. The cover may be taken to be finite

and we take K to be the intersection of all the Ki’s.

ButHn(G/K,M ′) = 0 asG/K is a finite p-group, so exists ψK : (G/K)n−1 →M such that φK = dnψK . Now set ψ : Gn−1 → (G/K)n−1 → M ′ so φ =dnψ.

Remark. The argument actually shows that if G = lim←−G/K then Hn(G,M) =

lim−→Hn(G/K,M).

Proposition 5.19. Let G be a pro-p group. The only simple p-primaryG-module is Fp.

Proof. Example sheet 4.

Combining these gives

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• 5 Cohomology of groups

Proposition 5.20. Let G be a pro-p group. If Hn(G,Fp) = 0 then Hn(G,M) =0 for all finite modules M .

Proposition 5.21. Suppose exists n such that Hn(G,M) = 0 for all Mthen cd(G) ≤ n− 1.

Non-examinable. By course convention we shall prove this for an abstract groupG. The main idea is dimension shifting: suppose there is a short exact sequenceof G-modules

0 M N M ′ 0

where N is cohomologically trivial. Then from the long exact sequence oneknow Hi(G,M ′) ∼= Hi+1(G,M).

Example sheet 4 shows that for K ≤ G, the following holds for the coinducedmodule coindKG (M) = HomZK(ZG,M):

Hn(G, coindKG (M))∼= Hn(K,M).

So take K = 1, have coindKG (M) = HomZ(ZG,M) cohomologically trivial.Finally the map

α :M → Hom(ZG,M)m 7→ (x 7→ xm)

is an injection so gives the required short exact sequence.

Corollary 5.22. Free pro-p groups have cohomological dimension 1.

Proof. Suffice to show H2(G,Fp) = 0, i.e. to show every extension of G by Fpsplits. Let G be free on X for some X finite. Suppose we have an extension

1 Fp E F (X) 1

E is again a pro-p group. For each x ∈ X choose a preimage ex ∈ E. It extendsto a unique homomorphism F (X)→ E.

Note that the proof works for free groups as well, so providing an algebraicproof of Proposition 5.1.

Theorem 5.23. Let G and G′ be tfg pro-p groups. Let f : G → G′ be acontinuous homomorphism. Assume

• f∗ : H1(G′,Fp)→ H1(G,Fp) is an isomorphism,

• f∗ : H2(G′,Fp)→ H2(G,Fp) is an injection.

Then f is an isomorphism.

Heuristics: H1(G,Fp) = Hom(G,Fp) = Hom(G/Φ(G),Fp), the dual of theFp-vector space G/Φ(G). The first condition tells us something about genera-tors. In particular f is a surjection. H2 is related to relations, and the secondcondition says that we impose no additional relations so f is an injection.

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• 5 Cohomology of groups

Non-examinable. Let Gn = γ(p)n (G), the lower central p-series. Recall that Gnare all open, G = lim←−G/Gn. Gn’s are fully characteristic, therefore inducingmaps fn : G/Gn → G′/G′n. We will show that they are all isomorphisms andso is f .

Induction on n. For n = 2, f2 : G/Φ(G)→ G′/Φ(G′). By the remark beforethe transpose of f2 is an isomorphism, so is f2 itself.

Suppose the result holds for n. If we can show Gn/Gn+1 → G′n/G′n+1 is anisomorphism then combining the induction hypothesis we deduce the result forn+ 1 from the diagram

1 Gn/Gn+1 G/Gn+1 G/Gn 1

1 G′n/G′n+1 G

′/G′n+1 G′/G′n 1

∼= ∼=

Gn/Gn+1 is a finite-dimensional Fp-vector space so Gn/Gn+1 → G′n/G′n+1 isan isomorphism if and only if its dual H1(G′n/G′n+1,Fp) → H1(Gn/Gn+1,Fp)is. A homomorphism φ : Gn → Fp factors through Gn/Gn+1 if and only ifφ([g, g′]) = 0 for all g ∈ G, g′ ∈ Gn, if and only if

0 = φ(g−1(g′)−1gg′) = −φ(g−1g′g) + φ(g′),

if and only if φ is G-invariant. Thus H1(Gn/Gn+1,Fp) = H1(Gn,Fp)G. Thefive term exact sequence induced by

1 Gn G G/Gn 1

says we have a commutative diagram of exact sequences

H1(G′/G′n) H1(G′) H1(G′n)

G′ H2(G′/G′n) H2(G′)

H1(G/Gn) H1(G) H1(Gn)

G H2(G/Gn) H2(G)

∼= ∼= ∼=

By induction hypothesis and injectivity on H2, the middle vertical map is anisomorphism by five lemma. Therefore Gn/Gn+1 ∼= G′n/G′n+1.

Theorem 5.24. If Γ and Γ′ are finitely generated abstract groups and f :Γ → Γ′ is a homomorphism and the same conditions hold, then f̂ : Γ̂(p) →Γ̂′(p) is an isomorphism.

Proof. Set Γn = γ(p)n (Γ). Then Γ̂(p) = lim←−Γ/Γn. Proceed as before.

Proof of Theorem 5.15. Suppose x1, . . . , xd is a generating set of minimal size.Let F be the free pro-p group on xi and consider f : F → G. F/Φ(F )→ G/Φ(G)is an isomorphism since they are the same Fp-vector space and H2(G,Fp) →H2(F,Fp) is an injecition. Thus f is an isomorphism.

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• 5 Cohomology of groups

Example. Let Γ = 〈x1, x2|x1x2x−11 x−12 x1〉. Recall H2(Γ,Z) = 0. The same

argument shows H2(Γ,Fp) = 0. H1(Γ,Fp) = Hom(Γ,Fp). Let φ : Γ → Fp.Then

0 = φ(x1x2x−11 x

−12 x1) = φ(x1)

and no further relation so H1(Γ,Fp) = Fp generated by x1 7→ 0, x2 7→ 1. Letf : Z→ Γ, 1 7→ x2. Then f induces Γ̂(p) ∼= Zp.

5.6.2 Presentation of pro-p groups

Definition (presentation of pro-p group). Let X be a finite set and F a freepro-p group on X. Let R ⊆ F . The pro-p group with presentation bX|Rcpis defined to be F/〈〈R〉〉.

Note that not all elements in bX|Rcp can be written as a product of elementsof X.

Lemma 5.25. Let Fabs be the abstract free group on X and let R ⊆ Fabs.Let Γ = 〈X|R〉, G = bX|Rcp. Then G = Γ̂(p).

Proof. Suffices to look at the p-group quotients of G and Γ. A quotient Γ→ P ,where P is a p-group, is the same as a function X → P such that its extensionto Fabs contains R in the kernel. But this is exactly the same as a continuousquotient G→ P .

Theorem 5.26. Let G be a tfg pro-p group. Let X be a finite tgs of G. LetrX be the minimal size of a set R
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