On the number of epi-, mono-, and homomorphisms of groups

TL;DR

This paper explores the divisibility properties of the number of group homomorphisms with specific conditions, extending known results to injective and surjective cases, with implications for counting generating pairs in finite groups.

Contribution

It generalizes classical divisibility results for homomorphism counts to include injective and surjective cases, providing new insights into the structure of homomorphisms under natural conditions.

Findings

Number of homomorphisms with certain properties is divisible by specific group invariants.

Derived a corollary about the number of generating pairs with element orders 3 and 5.

Established divisibility relations for homomorphisms satisfying natural conditions.

Abstract

It is known that the number of homomorphisms from a group $F$ to a group $G$ is divisible by the greatest common divisor of the order of $G$ and the exponent of $F/[F,F]$. We investigate the number of homomorphisms satisfying some natural conditions such as injectivity or surjectivity. The simplest nontrivial corollary of our results is the following fact: {\it in any finite group, the number of generating pairs $(x,y)$ such that $x^3=1=y^5$, is a multiple of the greatest common divisor of 15 and the order of the group $[G,G]\cdot\{g^{15}\;|\;g\in G\}$.