Another article on the number of homomorphisms

TL;DR

This paper extends the class of abelian groups satisfying the Asai-Yoshida conjecture on homomorphism counts and links divisibility problems in groups to this conjecture.

Contribution

It broadens the scope of the conjecture to more abelian groups and establishes a connection between divisibility issues and the conjecture.

Findings

Number of homomorphisms divisible by gcd(|G|, |F:F'|) under certain conditions

General result linking divisibility problems to the Asai-Yoshida conjecture

Extension of the conjecture to new classes of abelian groups

Abstract

We extend the class of abelian groups for which a conjecture of Asai and Yoshida on the number of crossed homomorphisms holds. We also prove a general result which connects certain problems concerning divisibility in groups to the Asai-Yoshida conjecture. One of the consequences is that for finite groups F and G the number |Hom(F,G)| is divisible by gcd(|G|, |F:F'|) if F/F' is a product of a cyclic group and a group with cube-free exponent.