ON THE COHOMOLOGY OF ALTERNATING AND SYMMETRICAL GROUPS AND DECOMPOSITION OF RELATION MODULES

[摘要] Let G = A(p), A(p+1), A(p+2), S(p), or S(p+1), where p greater-than-or-equal-to 5 is a prime. It is shown that if M is an irreducible G-module of characteristic q not-equal p, then dim H-1(G,M) less-than-or-equal-to (1/d)dim [P,M], where d = \N(G)(P):P\, P a Sylow p-subgroup of G. This is used to show that relation cores for these groups always decompose. This answers a question of Gruenberg and Roggenkamp, who had observed that the relation cores are indecomposable for the other alternating and symmetric groups. The result is also used to obtain bounds for the size of H-1(A(n),M) for all n.

[发布日期] 1990-12-31