
1. Introduction. Let S(X) denote the collection of homotopy classes of homotopy equivalences of a space X with itself. Composition of maps induces a group structure in S(X). From the point of view of categories S(X) is the group of equivalences of the object X in the category of spaces and homotopy classes of maps. Thus it is the homotopy analog of the automorphism group of a group and the group of homeomorphisms of a space.

In this note we present some theorems which relate properties of the homotopy groups of X to algebraic properties of S(X). In such a study one encounters the difficulties associated with the problem of composing homotopy classes of maps. In addition one can easily show that for any finite group T there exists a finite complex X such that &(X) contains T as a subgroup. Thus any group theoretic property which does not hold for all finite groups cannot be true for all groups 8>(X).

The hypotheses of all of our theorems are not intricate, and thus our results provide specific information on &(X) for many X. §2 contains theorems on the group of equivalences of any 1-connected finite complex, and §3 deals with associative iJ-spaces. At the end of each section we give a brief description of our methods. Details and applications will appear elsewhere.

Various results on &(X) have been obtained by Barcus-Barratt [l, §6], P. Olum (to appear) and D. W. Kahn (to appear). Furthermore, W. Shih [5] has constructed a spectral sequence for 8>(X).

We should like to thank R. P. Langlands for several discussions on Proposition 9.

2. General theorems. We consider only 1-connected spaces of the homotopy type of a CW-complex with finitely generated homotopy groups in all dimensions. Let X (n) be an nth Postnikov section of X. A straightforward obstruction argument yields LEMMA 1. If X is a finite CW-complex then 8(X) «S(X (w) ) for all w>dim X.

By assigning to an element of S(X) its induced homotopy automorphism, one obtains a homomorphism i": S(X)->JI* Aut iTk{X) whose kernel is denoted by S#(X). PROPOSITION 
There is an exact sequence where T n (X) denotes the kernel of the homomorphism H n (X; 7r n (X))
^Hom(7T n (Z),7r n (Z)).
There is an exact sequence where T n (X) denotes the kernel of the homomorphism H n (X; 7r n (X))
Note that the map T n (X)->8#(X (w) ) carries the sum of two cohomology classes into the composition of two homotopy equivalences. THEOREM 
Let X be a finite C\S/ J -complex such that rank Ti(X) ^1 for all igdim X + l. Then &(X) satisfies the maximal condition (i.e., S(X) and all its subgroups are finitely generated).
Examples of spaces X such that £>(X) does not satisfy the maximal condition can, for instance, be obtained from Corollary 8. Now let | T\ stand for the order of the group T and hi:Ti(X) ->Hi(X) for the Hurewicz homomorphism. As a group defined by composition of maps S(X) is generally nonabelian. Theorem 5 gives conditions for £(X) to be solvable. If p is a prime we denote by n(H, p k ) the number of times the cyclic group Zjjc occurs in the canonical decomposition of the finitely generated abelian group H. is dealt with purely algebraically. On the other hand it follows from Proposition 2 that &#(Y) possesses all properties which are shared by finitely generated abelian groups and which carry over to subgroups, factor groups and extensions. This observation immediately yields Theorem 3. The condition rank Ti(X) S1 of Theorem 4 implies that 8 (7) is finite if and only if 8# (7) REMARKS, (a) Clearly Theorems 3, 4 and 5 also hold for a space with finitely many homotopy groups.
Let X be a finite C\S/ J -complex such that rank Ti(X) ^1 for all igdim X + l. Then &(X) satisfies the maximal condition (i.e., S(X) and all its subgroups are finitely generated).
(b) It is easily seen that for a finite complex X the group 8#(X) is a subgroup of 8#(X (n) ) for #>dimX. Thus, for instance, 8#(X) always is solvable and satisfies the maximal condition.
Let X be a finite C\S/ J -complex such that rank Ti(X) ^1 for all igdim X + l. Then &(X) satisfies the maximal condition (i.e., S(X) and all its subgroups are finitely generated).
(c) There are examples of 1-connected finite complexes which show that each of the hypotheses in Theorems 3, 4 and 5 is necessary. The following theorem shows that for certain G the group 8(G) contains free subgroups of any rank. The proofs of Theorems 6 and 7 rely on the following considerations. First of all, the collection 7r(G, G) of homotopy classes of basepoint preserving maps from G into itself has the structure of a near-ring [2] . Secondly, it can be shown that the homomorphism 7r(G (n) , G (n) ) ->^fcgn Hom(x/c(G), iTk(G)) is an epimorphism modulo the class of finite abelian groups. In addition in the proof of Theorem 7 one needs the result of Frasch [4, §6] that the principal congruence group mod p of two-by-two integer matrices is a free group on at least two generators. An important step in the proof of Theorem 6(a) is the application of the following proposition to the ring TT(0\ G< tt) )/Ker 0 where 0: w(G™, G™)->r(QG™, QG<»>) is the loop homomorphism.
Let X be a finite C\S/ J -complex such that rank Ti(X) ^1 for all igdim X + l. Then &(X) satisfies the maximal condition (i.e., S(X) and all its subgroups are finitely generated).
PROPOSITION 9. Let A be an associative ring with 1 whose additive group is finitely generated. Then the group of units of A is also finitely generated.
Let X be a finite C\S/ J -complex such that rank Ti(X) ^1 for all igdim X + l. Then &(X) satisfies the maximal condition (i.e., S(X) and all its subgroups are finitely generated).
This proposition is a consequence of a result of Borel and HarishChandra [3, Theorem 6.12].

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