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

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 = p l , where p is prime and l ≥ 1 , n ≠ 2 , 4 , they conjectured a function C (p l) , that is polynomial in p , for the value of σ (p l). We prove that C (p l) is an upper bound for σ (p l). We introduce a function τ (n) for the number of abelianised groups and conjecture a function D (p l) , that is polynomial in p , for the value of τ (p l) , when p l ≠ 2 , 4 , 5 , 7 , 8 , 13 , 23. We prove that D (p l) is an upper bound for τ (p l). We pose three questions that ask if particular pairs of groups with common abelianisations are non-isomorphic. We prove that if τ (p l) = D (p l) and each of these questions has a positive answer then σ (p l) = C (p l). [ABSTRACT FROM AUTHOR]