A descent principle in modular subgroup arithmetic

[摘要] We establish and comment on a surprising relationship between the behaviour modulo a prime p of the number s(n) (G) of index n subgroups in a group G, and that of the corresponding subgroup numbers for a subnormal subgroup of p-power index in G. One of the applications of this result presented here concerns the explicit determination modulo p of s(n) (G) in the case when G is the fundamental group of a finite graph of finite p-groups. As another application, we extend one of the main results of the second author's paper (Forum Math, in press) concerning the p-patterns of free powers G*(q) of a finite group G with q a p-power to groups of the more general form H*G*(q), where H is any finite p-group. (c) 2005 Elsevier B.V. All rights reserved.

[发布日期] 2005-12-01