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We study the behaviour modulo p of the number sn(G) of index n subgroups in a group G, where p is an arbitrary prime, and G ranges over a substantial class Cp of free products depending on p. Among other things, we give an explicit combinatorial description of the mod p behaviour of sn(G) for G belonging to a large subclass of Cp, and we investigate the question under which conditions the set {n ∈ N : sn(G) 6≡ 0 (p)}, which captures much of the information contained in the mod p behaviour of sn(G), allows for a characterization in terms of closed formulae. Rather surprisingly, it turns out that Fermat primes play an important special role in the latter question, and, as a byproduct of our investigation, we obtain a new characterization of Fermat primes in terms of the subgroup arithmetic of Hecke groups.

Modular subgroup arithmetic in free products