Modular Subgroup Arithmetic and a Theorem of Philip Hall

A surprising relationship is established in this paper, 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 normal subgroup in G normal subgroup in p ‐power order. The proof relies, among other things, on a twisted version due to Philip Hall of Frobenius' theorem concerning the equation x m =1 in finite groups. 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 tree of groups all of whose vertex groups are cyclic of p ‐power order. Furthermore, a criterion is established (by a different technique) for the function S n G to be periodic modulo p . 2000 Mathematics Subject Classification 20E06, 20F99 (primary); 05A15, 05E99 (secondary).