Counting isomorphism classes of groups of Fibonacci type with a prime power number of generators

Abstract

Cavicchioli, O'Brien, and Spaggiari studied the number of isomorphism classes of irreducible groups of Fibonacci type as a function σ(n) of the number of generators n. In the case n=pl, where p is prime and l≥1, n≠2,4, they conjectured a function C(pl), that is polynomial in p, for the value of σ(pl). We prove that C(pl) is an upper bound for σ(pl). We introduce a function τ(n) for the number of abelianised groups and conjecture a function D(pl), that is polynomial in p, for the value of τ(pl), when pl≠2,4,5,7,8,13,23. We prove that D(pl) is an upper bound for τ(pl). We pose three questions that ask if particular pairs of groups with common abelianisations are non-isomorphic. We prove that if τ(pl)=D(pl) and each of these questions has a positive answer then σ(pl)=C(pl).