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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Some divisibility properties of the subgroup counting function for free products

Authors: Michael Grady and Morris Newman
Journal: Math. Comp. 58 (1992), 347-353
MSC: Primary 11B50; Secondary 20E06
MathSciNet review: 1106969
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Abstract: Let G be the free product of finitely many cyclic groups of prime order. Let $ {M_n}$ denote the number of subgroups of G of index n . Let $ {C_p}$ denote the cyclic group of order p , and $ C_p^k$ the free product of k cyclic groups of order p . We show that $ {M_n}$ is odd if $ C_2^4$ occurs as a factor in the free product decomposition of G . We also show that if $ C_3^3$ occurs as a factor in the free product decomposition of G and if $ {C_2}$ is either not present or occurs to an even power, then $ {M_n} \equiv 0\;\bmod\, 3$ if and only if $ n \equiv 2\;\bmod\, 4$ . If, on the other hand, $ C_3^3$ occurs as a factor, and $ {C_2}$ also occurs as a factor, but to an odd power, then all the $ {M_n}$ are $ \equiv 1\;\bmod\, 3$ . Several conjectures are stated.

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Article copyright: © Copyright 1992 American Mathematical Society

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