Abstract
Abstract. Previously nice vectorial equations were constructed having various finite classical groups as Galois groups. Here such equations are constructed for the remaining classical groups. The previous equations were genus zero equations. The present equations are strong genus zero.
Introduction
Let q > 1 be a power of a prime p, let k q be an overfield of GF(q), and let m > 0 be an integer. In previous papers [Ab4] , [Ab5] , [Ab6] over the fields k q (X), k q (X, T ), k q (X, T ) were shown to be the odd dimensional unitary, symplectic, and even dimensional negative (i.e., whose Witt index is one less than half its dimension) orthogonal groups SU(2m − 1, q ), Sp(2m, q), and Ω − (2m, q), respectively, where it was assumed that k q = k q (the algebraic closure of k q ); moreover, it was also assumed that, in the unitary case m > 1 and q = (q ) 2 for some power q > 1 of p, in the symplectic case m > 2, and in the orthogonal case m > 3 and p > 2. For the symplectic case, in [AL1] and [AL2] it was shown that the assumption k q = k q can be removed and the assumption m > 2 can be replaced by the assumption m > 1; moreover, in [In2] it was shown that after putting T = 1 in E (Y ), its Galois group over k q (X) remains Sp(2m, q).
Introduction
Likewise, in [Ab2] , [Ab3] , [Ab8] , [Ab9] , it was shown that the Galois groups of the vectorial trinomials In this paper we take care of the remaining classical groups by showing that the Galois groups of the following vectorial q-polynomials
Introduction
over k q (X) are the even dimensional unitary, odd dimensional orthogonal, and even dimensional positive (i.e, whose Witt index equals half its dimension) orthogonal groups U(2m, q ), SO(2m+1, q), and SO + (2m, q), where m > 1, m > 2, and m > 3, respectively; in the unitary case we assume q = (q ) 2 for some power q of p; in the odd dimensional orthogonal case when q is even we suppose m > 3 and note that SO equals O; in the even dimensional orthogonal case when q is even the answer may be Ω + (2m, q). According to the terminology introduced in [Ab7] , the symplectic and special linear equations, i.e., the polynomials Y −1 E (Y ) and Y −1 E * (Y ), are genus zero, i.e., they are linear in X, the odd dimensional unitary and even dimensional negative orthogonal equations, i.e., the polynomials Y −1 E † and Y −1 E − (Y ), are almost genus zero, i.e., they have genus zero factors having the same splitting fields as the polynomials themselves, the general linear equation, i.e., the polynomial Y −1 E * * (Y ), is strong genus zero, i.e., it is genus zero and in it only the constant term involves X, and the new polynomials
Introduction
, and Y −1 E + (Y ) will be shown to be almost strong genus zero, i.e., they have strong genus zero factors having the same splitting fields as the polynomials themselves. Again as said in [Ab7] , this is in concordance with the Guralnick-Saxl List given in Theorem 3.1(B) of their paper [GSa] . Indeed it was this list which prompted the construction of the last three polynomials. We shall explain the link between our approach and that of Guralnick and Saxl in the remarks at the end of the paper.
Introduction
The establishment of the Galois groups of the first three polynomials E † , E , E − , was based on the Orbit Size Theorems of Liebeck [Li1] , the Rank 3 Theorems of Kantor [Kan] , and the Antiflag Transitive Theorems of Cameron-Kantor [CKa] , and involved some very intricate polynomial factorizations developed for applying these Recognition Theorems of Group Theory. At the end of [Ab7] these factorizations were codified into a MANTRA. In [AL1] , [AL2] , and [In2] , this mantra helped to convert the subdegree factorizations into factorizations yielding the underlying symplectic forms. Likewise, in this paper we shall use the mantra to construct the relevant quadratic and hermitian forms and hence to factorize the original polynomials.
Introduction
By writing (E † † , q, m) for (E ‡ , q , 2m) we get the vectorial q-polynomial
Introduction
where m > 2 (and q need not be a square). This is the incarnation of E ‡ we shall mostly deal with, and we shall show that in case of odd m the Galois group of E † † over k q (X) is SO + (2m, q) except that when q is even it may be Ω + (2m, q). For even m, as indicated, by changing notation we revert to E ‡ and, as said above, if GF(q 2 ) ⊂ k q , then the Galois group of E † † over k q (X) is U(m, q).
Introduction
In Sections 2-5 we shall discuss some generalities about quadratic and hermitian forms. Then in Section 6, as our main step, by an implicit invocation of the Mantra, we shall factor the three vectorial q-polynomials E † † , E
Introduction
• , E + , and show that each one of them gives rise to a quadratic form on its root-space which is left invariant by the corresponding Galois group. Finally, in Section 7 we shall calculate the Galois groups; there we shall also review the definitions of the orthogonal and unitary groups.
Introduction
Let us start off by recalling that, according to [Ab8] , a separable vectorial qpolynomial of q-degree d (with integer d ≥ 0) over an overfield K of k q is a polynomial of the form
Introduction
with a i ∈ K and a 0 = 0 = a d (where separable refers to a d = 0 and monic means a 0 = 1) and upon letting V [E] = (the root-space of E) = the set of all roots of E in K where K is the algebraic closure of K, we have the following. (Observe that we are using ≤ to denote subgroup, although the first author used < in his previous papers.)
Introduction
, which by choosing a basis of V [E] may be identified with the group GL(m, q) of all m × m nonsingular matrices over GF(q).
Introduction
For any finite-dimensional GF(q)-subspace V of K we put
Introduction
and then, according to the partial converse of (1.1) proved in (3.9) of [Ab9] , we see that:
It follows that:
Note (1.3). V → f V gives a bijection of the set of all finite-dimensional k-subspaces of K onto the set of all monic separable vectorial q-polynomials over K; in this bijection, a d-dimensional V corresponds to a vectorial q-polynomial of q-degree d; in particular, the zero space corresponds to the polynomial Y .
It follows that:
Abhyankar Conjecture) says that for any nonnegative integer t and any nonsingular projective curve C g of genus g over an algebraically closed ground field of characteristic p we have π A (C g,t ) = Q 2g+t (p) where the algebraic fundamental group π A (C g,t ) of C g,t = C g minus t + 1 points is defined to be the set of all Galois groups of unramified finite Galois coverings of C g,t , and where Q t (p) is the set of all finite groups G for which G/p(G) is generated by t generators with p(G) being the subgroup of G generated by all of its p-Sylow subgroups. The brilliant proof of this given by Harbater [Har] and Raynaud [Ray] being existential in nature, it is worthwhile to continue with the project of finding explicit equations for the said coverings. Moreover, explicit equations tend to enhance our knowledge of what happens over finite ground fields.
Quadratic forms
As common notation for the next four sections, let V be a d-dimensional (dfinite) vector space over a field k, and let e 1 , . . . , e d be a basis of V . Let Lin(V ) be the k-vector-space of all linear functions α :
Quadratic forms
gives an isomorphism of Lin(V ) onto the k-vector-space of all homogeneous linear polynomials in k [X] , where h α is characterized by the condition
Quadratic forms
Let Bil(V ) be the k-vector-space of all bilinear functions b :
Quadratic forms
gives an isomorphism of Bil(V ) onto the k-vector-space of all homogeneous bilinear polynomials in k[X, X ], where h b is characterized by the condition
Quadratic forms
Next we claim "product generation" which says that Bil(V ) is generated by "product elements," i.e., elements of the form γ :
Quadratic forms
for all u, v in V , with α and β in Lin(V ). Note that such a product element, which we may denote by αβ, clearly belongs to Bil(V ). To prove the claim, given any
Quadratic forms
is given by γ ij (e i , e j ) = δ ii δ jj in terms of the Kronecker delta. Let α i in Lin(V ) be given by α i (e i ) = δ ii . Then γ ij (e i , e j ) = α i (e i )α j (e j ). Therefore γ ij equals the product element α i α j .
Quadratic forms
Let Qua(V ) be the k-vector-space of all quadratic functions on V . Recall that these are functions c : V → k such that c(λv) = λ 2 c(v) for all λ ∈ k and v ∈ V and such that the associated bivariate function b (u, v) 
Quadratic forms
gives an isomorphism of Qua(V ) onto the k-vector-space of all homogeneous quadratic polynomials in k [X] , where h c is characterized by the condition c(
Quadratic forms
is a homogeneous quadratic polynomial, then upon letting
Quadratic forms
we always have "bilinear generation"
Quadratic forms
and for odd characteristic we have "symmetric generation"
Quadratic forms
Note that h * is symmetric and is independent of the choice of basis whereas h need not be symmetric and is (highly) basis dependent.
Quadratic forms
By the isometry group of c ∈ Qua(V ) we mean the subgroup of GL(V ) consisting of all τ ∈ GL(V ) such that c(τ (v)) = c(v) for all v ∈ V . Note that the bilinear form b(u, v) = c(u + v) − c(u) − c(v) associated with c ∈ Qua(V ) is obviously symmetric. We define the kernel of a symmetric bilinear form b on V to be the subspace ker b = {u ∈ V : b(u, v) = for all v ∈ V } of V . For any c ∈ Qua(V ), upon letting b be the symmetric bilinear form on V associated with c, we define the kernel of c to be the subspace ker c = {u ∈ ker b : c(u) = 0} of ker b; the quadratic form c is said to be degenerate or singular according to whether ker b = 0 or ker c = 0.
Quadratic forms
In older literature, homogeneous linear polynomials (resp., homogeneous bilinear polynomials, homogeneous quadratic polynomials
Quadratic forms
were called linear forms (resp., bilinear forms, quadratic forms); following current convention, their basis free incarnations, i.e., members of Lin(V ) (resp., Bil(V ), Qua(V )) may be called linear forms (resp., bilinear forms, quadratic forms) on V , and the polynomials themselves may be called linear multiforms (resp., bilinear multiforms, quadratic multiforms).
Hermitian forms
Given an automorphism σ of k, let k be the fixed field of σ. For any function α : V → k let σα : V → k be the function such that (σα)(v) = σ(α(v)) for all v ∈ V , and for any polynomial h in one or more variables with coefficients in k let σh be the polynomial in the same variable or variables obtained by applying σ to the coefficients of h. Let Sem(V ) be the k-vector-space of all semilinear (with respect to σ) functions α : V → k, i.e., additive functions α : V → k such that for all λ ∈ k and v ∈ V we have α(λv) = σ(λ)(v); members of Sem(V ) may be called semilinear forms on V . Note that then
Hermitian forms
gives an isomorphism of Sem(V ) onto the k-vector-space of all linear multiforms in k [X] , where h α is characterized by the condition
Hermitian forms
Let Ses(V ) be the k-vector-space of all sesquilinear functions b : V × V → k, i.e., functions which are linear in the first variable and semilinear in the second variable; members of Ses(V ) may be called sesquilinear forms on V . Note that then
Hermitian forms
gives an isomorphism of Ses(V ) onto the k-vector-space of all bilinear multiforms in k[X, X ], where h b is characterized by the condition
Hermitian forms
Henceforth, assume that σ is of order two. Note that then α → σα gives a k -linear isomorphism of Lin(V ) onto Sem(V ) as well as a k -linear isomorphism of Sem(V ) onto Lin(V ). Also note that, for any α ∈ Lin(V ) ∪ Sem(V ) we have h σα = σh α . Finally, as in the previous section, we get sesquiproduct generation saying that, as a k-vector-space, Ses(V ) is k-generated by sesquiproduct elements 
Hermitian forms
For hermitian functions b ∈ Her(V ) we have "diagonal determination," i.e., b(u, v) is determined by its diagonal values b (v, v) ; to see this, it suffices to show
Hermitian forms
and taking λ = 1 we get
Hermitian forms
and hence for any λ ∈ k,
Hermitian forms
and now taking λ ∈ k \ k we conclude that b(u, v) = 0. Actually this only settles the uniqueness part of "diagonal determination." In Note (3.1), we shall discuss the existence part for finite fields.
Hermitian forms
In the rest of this section assume that k = GF(q) where q is a square. Then upon letting q = q 1/2 we have k = GF(q ), and for all λ ∈ k we have σ(
Hermitian forms
Note (3.1). Given any hermitian form b on V , let c(u) = b(u, u) for all u ∈ V . Now b is a bilinear form on V considered as a vector space over k and so c is a quadratic form on V considered over k . Moreover,
Hermitian forms
for all λ ∈ k and u ∈ V . This is precisely the condition for a quadratic form over k to arise from a hermitian form over k. Indeed, if c is a quadratic form over V considered over k such that for all λ ∈ k and u ∈ V we have c(λu
Hermitian forms
Namely, ( * ) can be proved by frequent use of the following simple fact about finite fields: 
Hermitian forms
It follows that b is bilinear over k . Now for β ∈ k × and x, y ∈ V we have
Hermitian forms
and so b is linear (over k) in the first variable. Next, for any x, y ∈ V we have
Hermitian forms
q so that b is hermitian-symmetric and hence semilinear in the second variable. Finally, for any x ∈ V we have
Hermitian forms
This completes the proof of (3.1).
Hermitian forms
Next we note that
Hermitian forms
gives an isomorphism of Sem(V ) onto the k-vector-space of all protosemilinear multiforms in k [X] , where h α is characterized by the condition
Hermitian forms
and where by a protosemilinear multiform we mean a polynomial
Hermitian forms
gives an isomorphism of Ses(V ) onto the k-vector-space of all protosesquilinear multiforms in k [X, Z] , where h b is characterized by the condition
Hermitian forms
and where by a protosesquilinear multiform we mean a polynomial
Quadratic polynomials
Now assume that k = GF(q) where q need not be a square, and note that then
Quadratic polynomials
, and d > 0. Henceforth, we may use (1.1) to (1.3) tacitly, and we may use q-linear polynomial as a synonym for vectorial q-polynomial.
Quadratic polynomials
Given any 0 = α ∈ Lin(V ), upon letting U = ker α we see that U is (d − 1)-dimensional, and hence by linearity α(w)/f U (w) is independent of w ∈ V \ U , and so we may define
Quadratic polynomials
In view of product generation, it follows that, given any b ∈ Bil(V ), there exists
Quadratic polynomials
where by q-bilinear we mean g b is of the form
Quadratic polynomials
Now in view of bilinear generation we see that, given any quadratic function c on
Quadratic polynomials
where by q-quadratic we mean g c is of the form
Quadratic polynomials
Note (4.1). Note that the uniqueness of g c follows by
Hermitian polynomials
Let the situation be as in the previous section, and assume that q is a square. Let q = q 1/2 , and let σ be the order-two automorphism of k given by σ(λ) = λ q for all λ ∈ k. Note that now the fixed field of σ is k = GF(q ). From (3.1) and Section 4, we see that given any b ∈ Her(V ), there exists a unique
Hermitian polynomials
where
Hermitian polynomials
have the exhibited expressions, and
Hermitian polynomials
for all u ∈ V and λ ∈ k, and hence
Hermitian polynomials
Note (5.1). Note that, in view of diagonal determination, the uniqueness of g b also follows by observing
K); this follows by observing that g(v) = t( g(v)) = (t g)((t(v)) where (t g)(Y ) ∈ K(V )[Y ] is obtained by applying t to the coefficients of g(Y ). Finally, note that for any g(Y
) ∈ K[Y ] we have: g(V ) ⊂ k ⇔ [ g q (Y ) − g(Y )]/f V (Y ) ∈ K[Y ].
Forms and factorizations
We shall now apply (4.1) to find quadratic forms associated with the three vectorial polynomials
Forms and factorizations
Note that each term in the above summations evaluated at any w ∈ k equals w 2 , and the total number of terms in the two summations involved in J
Forms and factorizations
• is the odd integer 2m − 1, and hence (1*) if q is even, then J
Forms and factorizations
• (w) = w 2 for all w ∈ k. Now we have the following Quadratic Form Theorem. 
Quadratic Form Theorem (6.1). Let us put
(E, A, I, J, C) = (E † † , A † † , I † † , J † † , C † † ) or (E • , A • , I • , J • , C • ) or (E + , A + , I + , J + , C + ) with d = 2m ≥ 6 or d = 2m+1 ≥ 7 or d = 2m ≥ 8 respectively. Then clearly A(Y ) ∈ k[Y ] is separable monic q-linear of q-degree d − 2 such that if d is odd, then A(Y ) = Y q d−2 + Y , I(Y ) = Y q (d−2)/2 or Y q (d−1)/2 − Y q (d−3C(Y ) = X −1 A(Y ) q+1 + J(Y ) q (6.1.1) with A(Y )I(Y ) = J(Y ) q − J(Y ) (6.1.2) and E(Y ) = A(Y ) q 2 − X q−1 A(Y ) + X q I(Y ) q (6.1.3) and hence C(Y ) ∈ K[Y ] is q-quadratic of q-quadegree ≤ d − 1 with X −q A(Y ) q E(Y ) = C(Y ) q − C(Y
To establish the above claim let B(Y, Z)
q with A(w) = 0 and J(w) = w 2 , and hence c(w) = w 2 . We shall now prove the following Factorization Theorem where, as notation, for any nonzero P (Y ) and
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
; note that this is independent of the field K as long as it contains the coefficients of P (Y ) and Q(Y ); also note that a monic polynomial is a nonzero polynomial in which the highest degree coefficient is 1. As a preparation for the said theorem, we note that
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
, and by peeling off the initial term in the first summation involved in J • and then suitably pairing the remaining terms in the two summations we get
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
and hence: (2*) I
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
Note that k × is the set of all nonzero elements of k, and k ×2 is the set of all squares in k × ; also note that if q is even, then every w ∈ k has a unique square root in k which we denote by √ w. Now by expanding the first two terms of J • (Y ) in (2*) around a value in k and noting that the remaining terms are divisible by very high powers of (Y q − Y ), we see that: (4*) for any w ∈ k × \ k ×2 (i.e., for any nonzero nonsquare w in k) we have GCD(I • (Y ), J • (Y ) − w) = 1; and (5*) if q is odd, then for any
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
; and: (6*) if q is even, then for any w ∈ k × we have
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
We can alternatively deduce the above by noting that I Factorization Theorem (6.2). In the situation of (6.1), for every w ∈ k let
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
and the splitting field of E w (y) over K coincides with the splitting field
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
and, noting that
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
, and letting
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
where
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
we have
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
where
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
and finally, we have 
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
To prove this, for every w ∈ k let
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
Then obviously
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
In view of the well-known fact that, in dimension greater than 2, the orthogonal group acts faithfully on each of its orbits on nonzero vectors, we see that, for every For every w ∈ k, since w q = w, by (6.1.1) we also see that
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
, and therefore upon letting
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
we have
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
and by the Gauss Lemma we see that 
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
whereas if m is even and we are in the two-dagger case, 
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
(6.2.14)
by GCD(P (Y ), Q(Y )) we denote the monic generator of the ideal generated by P (Y ) and Q(Y ) in K[Y ]
Now for a moment assume that d = 2m+1 is odd and q is odd with w ∈ k × \k ×2 . Then by (4*) we see that GCD(I(Y ), J(Y ) − w) = 1. Therefore, by (6.1.2) and (6. 
0). Upon letting A(Y ) = A(Y )/Y we have A(Y ) ∈ k[Y ] with A(Y ) = Y A(Y ), and upon letting
P w (Y ) = A(Y ) q+1 /(J(Y ) − w) q we have P w (Y ) ∈ k[Y ] with P w (Y ) = Y q+1 P w (
Y ). Now A(Y ) and J(Y ) − w belong to k[Y 2 ] and hence so does P w (Y ) and therefore
and
Y ). Now A(Y ) and J(Y ) − w belong to k[Y 2 ] and hence so does P w (Y ) and therefore
(6.2.15)
Y ). Now A(Y ) and J(Y ) − w belong to k[Y 2 ] and hence so does P w (Y ) and therefore
Finally, for a moment assume that d = 2m + 1 is odd and q is even with w ∈ k × . Then w has a unique square root in k × , and hence by (6*) we see This completes the proof of (6.2).
Galois groups
Again let
Galois groups
where in a moment we shall assume that K = k q (X). We shall now calculate the Galois groups of the vectorial polynomials
Galois groups
To do this our polynomial manipulation material will be Theorems (6.1) and (6.2) of Section 6, and our group theory tools will be Liebeck's Theorem on orbit sizes of classical groups [Li2] and the Guralnick-Saxl Theorem on Strong Genus Zero Polynomials [GSa] . We shall also use Hering's Theorem as given by Liebeck in the Appendix of [Li1] , and the criterion for the Galois group of a vectorial polynomial to be contained in SL or Ω given in [In1] .
Galois groups
To 
Galois groups
2 where q is a power of p, then, given a nondegenerate hermitian form b on V , by U (V, b) we denote the isometry group of b, and we put SU(V, b) = U(V, b) ∩ SL(V ). Moreover, we put ΓU(V, b) = the subgroup of ΓL(V ) consisting of those g ∈ ΓL(V ) for which there exists λ ∈ k × and α ∈ Aut(k) such that b (g(u), g(v)) = λα(b(u, v) ) for all u, v in V , and we put GU(V, b) = the subgroup of GL(V ) consisting of those g ∈ GL(V ) for which there exists 
Galois groups
Given a vector space V over GF(q 2 ) of finite dimension m > 0, in a natural manner V can be regarded as a d-dimensional vector space V over k with d = 2m, and we call this V the cognate of V , and we note that if we let Φ = HL( V ) ∪ {0} ∼ = GF(q 2 ), then we have the normalizer and centralizer equa-
Galois groups
Moreover, given any nondegenerate hermitian form b on V , there is a unique nonsingular quadratic form c on V such that c(v) = b(v, v) for all v ∈ V , and we call this c the cognate of b. Conversely, given any vector space V over k of even finite dimension d = 2m > 0, in several ways we may regard V as an m-dimensional vector space V over GF(q 2 ) and we call such a V an antecognate of V , and we note that then V is the cognate of V . Also note that now a nonsingular quadratic form c on V is the cognate of at most one nondegenerate hermitian form b on V , and when such a b exists we call it a V -antecognate of c. , and now by (1.1) in a natural manner we have
Galois groups
In his Orbit Size Theorem [Li2] , Liebeck discusses subgroups of PGL and the orbit sizes of their action on the projective space, whereas our Theorem (6.2) provides information about subgroups of GL and the orbit sizes of their action on the vector space. Here are the relevant portions of the two versions of the Orbit Size Theorem, first extracts from Liebeck's original projective version and then the vectorial version in the form we need. In converting the projective to the vectorial version we shall use some material in Section 5 of [Ab4] and Section 5 of [Ab6] . Note that cases 
Vectorial Orbit Size Theorem (7.2). Given any monic separable vectorial q-
and (E, K) ) has a normal subgroup isomorphic to Ω(7, q). 
then witt(c) = m and Ω(V [E], c) Gal(E, K). (7.2.d) If d = 2m + 1 ≥ 7 is odd with q odd, and
For deducing (7.2) from (7.1), we need the following three auxiliary lemmas.
then witt(c) = m and Ω(V [E], c) Gal(E, K). (7.2.d) If d = 2m + 1 ≥ 7 is odd with q odd, and
Unitary Lemma (7.3). Let V be a vector space over k of finite dimension d > 2.
Assume that q = (q )
2 where q is a power of p, and let b be a nondegenerate hermitian form on V . Then we have the following.
Assume that q = (q )
(7.3.1) For any H ≤ GL(V ) we have
Assume that q = (q )
The four implications of (7.3.1) are proved in (5.1), (5.6), (5.2) and (5.4) of [Ab4] respectively. To prove (7. PGO(V, c) , PGO(V, c) .
Assume that q = (q )
The four implications of (7.4.1) are proved in (5.1), (5.6), (5.2) and (5.4) of [Ab6] respectively; although in [Ab6] , d was assumed even, the argument given there applies when d is odd. To prove (7. V , b) ). The proof of the first implication of (7.5.1) is completely parallel to the proofs of the first implications of (7.3.1) and (7.4.1). They all follow from the following claim: Let Θ : L → L be an epimorphism of finite groups such that ker Θ is contained in the center Z(L) of L, and (a given prime) p does not divide its size |ker Θ|. Also let S be a quasi-p subgroup of L (i.e., S is generated by p-power order elements), and let H be any subgroup of L.
Assume that q = (q )
To apply the claim take Θ V and L = GL(V ), and in cases (7.3.1), (7.4.1), (7.5.1) take S = SU (V, b) , Ω(V, c), SU( V , b) respectively. Also note that the quasi-p-ness of SU(V, b), and hence of SU( V , b), is indicated near (5.1) of [Ab4] , and the quasip-ness of Ω(V, c) is indicated near (5.1) of [Ab6] .
Assume that q = (q )
The "⇒" parts of the claim are obvious. Now suppose Θ(S) ≤ Θ(H). Take any p-power order element π in S. Then Θ(π) is p-power order. Also Θ(π) ∈ Θ(S) ≤ Θ(H) and hence Θ(π) = Θ(π ) for some π ∈ H. Since ker Θ ≤ Z(L) and |ker Θ| is nondivisible by p, we get π = απ for some α ∈ Z(L) such that the order µ of α is nondivisible by p. Now π µ = π µ and hence upon letting x denote subgroup generated by x we get π ∈ π = π µ = π µ ≤ H. Since S is generated by such elements π, we conclude that S ≤ H. Now suppose that Θ(S) Θ(H). Then for every π as above, and every h ∈ H we have that Θ(h)Θ(π)Θ(h −1 ) is a p-power order element in S; taking a Θ-preimage π * of this in S we have hπh −1 = βπ * for some β ∈ ker Θ; Since the order ν of β is prime to p, we get hπ
Assume that q = (q )
This being so for every such π, we conclude that S H.
Assume that q = (q )
The proof of the second implication is also completely parallel to the proofs of the second implications of (7.3.1) and (7.4.1). They all follow from the obvious facts that for any group epimorphism Θ : L → L and any subgroups H and
Assume that q = (q )
As in (7.3.1) and (7.4.1), our third implication is equivalent to the equation
Assume that q = (q )
To prove this normalizer equation, let C L (S) be the centralizer of S in L, and let us consider the endomorphism ring R of V over GF(q); note that R is naturally isomorphic to the ring of 2m by 2m matrices over GF(q), and L is the set of all units in R. By (2.10.2) on page 48 of [KLi] we see that C L (S) is the multiplicative group of all nonzero elements in an overfield Φ of Ψ = HL(V ) ∪ {0} in R whose field degree δ is a factor of 2m. Clearly, HL( V ) ∪ {0} is a subfield of Φ and its field degree over Ψ is 2, and hence δ = 2 for some integer > 0. Now Φ is isomorphic to GF(q 2 ) and V can be regarded as an (m/ )-dimensional vector space V over GF(q 2 ) and then we get S ≤ GL( V ) 
Assume that q = (q )
with h ∈ GL( V ) and i = 0 or 1. Moreover, gSg
Assume that q = (q )
hence by the third implication of (7.3.1) we see that
Assume that q = (q )
The fourth implication follows from the third exactly as in (7.3.1) and (7.4.1). Alternatively, it also follows from the second half of the claim and the first three implications.
Assume that q = (q )
To prove (7.5.2), let w ∈ k × and H ≤ ΓL( V ) be such that every h ∈ H maps , c) , we see that ( F w ) w∈k is a subpartition of (F w ) w∈k , i.e., each set F w is the union of a certain number of sets F w . Therefore, by the possible sizes of F w and F w (which are well-known), we conclude that F 0 = F 0 and F w = F σ(w) for all w ∈ k × where σ is a permutation of k × . Now by (7.5.2) we see that, for every w ∈ k × , H acts imprimitively on F w = {v ∈ V \ {0} : c(v) = w} with blocks of size q + 1.
Assume that q = (q )
This completes the proof of (7.5).
Assume that q = (q )
Remarks.
Assume that q = (q )
(1) Note that in the three cases, i.e., in (7.3.1), (7.4.1), and (7.5.1), the second part of the claim (which we have proved directly) follows from the four implications of each case.
Assume that q = (q )
(2) An alternative approach to the first and the fourth implications of (7.5.1) would proceed via the following easy result. Let L be a finite group and M a normal subgroup of L contained in the center Z(L) of L, and let S be an unsolvable subgroup of L with the property that all proper normal subgroups of S are contained in the center Z(S) of S. Then for any subgroup H of L we have
Assume that q = (q )
(3) The second part of the proof of the third implication of (7.5.1) which involves some explicit constructions may be replaced by a more conceptual argument. Namely,
Assume that q = (q )
It only remains to show that N ΓL2( V ) (S) = ΓU 2 ( V ) which is straightforward. Now we turn to deducing (7.2) from (7.1). (SU( V , b ) ) G for some nondegenerate form b on V , and then by (7.3) we get SU( V , b) H.
Assume that q = (q )
Next, to prove (7.2.b), assume that d = 2m ≥ 8 is even with m even, and
.2) If q is odd and c is a nonsingular quadratic form on
Remark. We take this opportunity to point out that a statement in [In1] is incorrect: "G is a subgroup of the kernel of the spinor norm if and only if c is a nonsquare in F q " should read "G is a subgroup of the kernel of the spinor norm if and only if c is a square in F q " and the index d/2 in the definition of c α appears as d.
.2) If q is odd and c is a nonsingular quadratic form on
We shall now prove the following corollary of (7.6.1), (7.6.2), (6.2.12), and (5.1).
Corollary (7.7).
In the situation of (6.1) with K = k q (X) Namely, (7.7.1) follows from (7.6.1) by noting that, by letting a be the coefficient of Y in E(Y ), we have (−1) d a = X q−1 = a (q − 1)-th power in K. Likewise, if q is odd, then (7.7.2) follows from (7.6.2) by noting that, if d is even, then, in view of (6.2.12), for every w ∈ k × we have (−1) Dw/2 E w (0) = ±X ∈ k ×2 , and if d is odd, then, in view of (6.2.15), for every w ∈ k × \ k ×2 we have (−1) Dw/2 E w (0) = ±X ∈ k ×2 . To prove (7.7.3) assume that d = 2m ≥ 6 is even and m is even with E = E + , and suppose if possible that the action of Gal(E, K) on F 1 = {v ∈ V [E] : c(v) = 1} is imprimitive with blocks of size q + 1. By (6.2.4), (6.2.6), (6.2.7), (6.2.12), the action of Gal(E, K) is faithful on the roots F 1 of E 1 (Y ) = P 1 (Y ) + X where m a = X q−1 , the largest divisor l of q 2 − 1 for which this is an l-th power in K is q − 1. Consequently, by (7.6.1) we see that the image of Gal(E, K) under the determinant map of GL( V [E]) onto the nonzero elements of GF(q 2 ) has order q + 1. Therefore, where the last equality is well-known.
Corollary (7.7).
Finally, we are ready to prove our Main Theorem.
Corollary (7.7).
Main Theorem (7.8). In the situation of (6.1) with K = k q (X), by letting c be the nonsingular quadratic form on V [E] given by v → C(v), we have the following.

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