Modular Subgroup Arithmetic and a Theorem of Philip Hall |
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Authors: | Muller Thomas W |
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Institution: | School of Mathematical Sciences, Queen Mary and Westfield College Mile End Road, London E1 4NS t.w.muller{at}qmul.ac.uk |
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Abstract: | A surprising relationship is established in this paper, betweenthe behaviour modulo a prime p of the number Sn G of index nsubgroups in a group G, and that of the corresponding subgroupnumbers for a normal subgroup in G normal subgroup in p-powerorder. The proof relies, among other things, on a twisted versiondue to Philip Hall of Frobenius' theorem concerning the equationxm=1 in finite groups. One of the applications of this result,presented here, concerns the explicit determination modulo pof Sn G in the case when G is the fundamental group of a treeof groups all of whose vertex groups are cyclic of p-power order.Furthermore, a criterion is established (by a different technique)for the function Sn G to be periodic modulo p. 2000 MathematicsSubject Classification 20E06, 20F99 (primary); 05A15, 05E99(secondary). |
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