A Descent Principle in Modular Subgroup Arithmetic

We establish and comment on a surprising relationship between the behaviour modulo a prime p of the number sn(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 sn(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 [16] 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.