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Modular Subgroup Arithmetic and a Theorem of Philip Hall
Author(s) -
Müller Thomas W.
Publication year - 2002
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609302001212
Subject(s) - mathematics , modulo , normal subgroup , combinatorics , prime power , vertex (graph theory) , modular group , order (exchange) , prime (order theory) , prime number , group (periodic table) , discrete mathematics , graph , chemistry , organic chemistry , finance , economics
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).