Another article on the number of homomorphisms
Alexander V. Khudyakov
TL;DR
This work extends the Asai–Yoshida framework by proving divisibility results for crossed homomorphisms in a broad class of $M$-indexed groups, linking these counts to the conjecture via a tail/core decomposition. The authors develop the tail- and core-based machinery and establish a general theorem that connects divisibility in crossed-homomorphism counts to the Asai–Yoshida conjecture. They prove the restricted version of the main conjecture for $M = \mathbb{Z}/p^n\mathbb{Z} \times (\mathbb{Z}/p\mathbb{Z})^m \times (\mathbb{Z}/p^2\mathbb{Z})^k$ and derive corollaries that yield divisibility of $|\mathrm{Hom}(F,G)|$ by $\gcd(|G|,|F:F'|)$ when $F/F'$ has the described structure, including reductions to $p$-groups and implications for abelian and certain non-abelian targets. The approach unifies several known cases and provides a systematic method to deduce global divisibility statements from crossed-homomorphism counts, with potential applications to group enumeration and structural counting problems.
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.