On Exponents of Homology and Cohomology of Finite Groups

Let G be a finite group and let r be the maximum of the p-ranks of G for all primes p dividing the order to G. There exist positive integers m and n such that the exponents of Hn(G,Z) and Hm(G, Z) are greater than Suppose that G is a finite group. It has long been known that the exponent of Hn(G, Z) divides |G|, the order of G, for all n > 0. Beyond this fact very little has been established concerning exponents of cohomology groups, and what is known seems to be mostly about upper bounds for the exponents. Lewis [3] found an element of order p2 in H2p{G, Z) when G is the group of order p3 and exponent p (p odd). The purpose of this paper is to give a general lower bound for exponents on group homology and cohomology. In some sense it explains the phenomenon observed by Lewis. Throughout let R denote either the ordinary integers Z or the localization Zp of Z at a prime p. Define the i?-rank of G to be equal to the p-rank of G if R = Zv and equal to the maximum of the p-ranks of G for all primes p if R = Z. The main result is the following. THEOREM 1. Let r be the R-rank of G. There exist an infinite number of positive integers m and n such that expHn(G,R)>\R/(g)\1/r and expHm{G,R) >\R/{g)\1/r, where \R/(g)\ is the order of the group R/(g), g = |G|. In the case R = Z, \R/{g)\ = |G|. If .R = Zp and |G| = pa-q, then R/{g) has order pa. It is well known that, because Zp is a flat Z-module, Hn{G, Z) = 0p Hn{G, Zp) and Hm{G,Z) = 0p#m(G, Zp) for m,n > 1. That is, Hn(G,Zp) is isomorphic to Hn(G, Z)p, the p-primary part of Hn(G, Z). Hence in case R = Zp the theorem implies the following. COROLLARY 2. Suppose that \G\ = pa ■ q, (p,q) = 1, and a > 1. Let r be the p-rank of G, and let s be the least integer that is greater than or equal to a/r. Then there exist an infinite number of positive integers m and n such that ps divides expHn{G,Z) andexpHm{G,Z). The proof of Theorem 1 is little more than a combination of two results. The first demonstrates that certain types of complexes of projective i?G-modules can Received by the editors December 8, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 20J06.