Some divisibility properties of the subgroup counting function for free products

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.