Divisibility relation between the number of certain surjective group and ring homomorphisms

TL;DR

This paper explores a divisibility relationship between the counts of ring and surjective group homomorphisms for finite cyclic and abelian structures, revealing new algebraic divisibility properties.

Contribution

It establishes a novel divisibility relation between the number of ring homomorphisms and surjective group homomorphisms in finite cyclic and abelian groups.

Findings

Number of ring homomorphisms divides the number of surjective group homomorphisms for certain cyclic groups.

The result applies when the codomain order n is not of a specific form involving primes congruent to 3 mod 4.

Extension of the divisibility relation to finite abelian groups.

Abstract

In this article, we identify the existence of a divisibility relationship between the number of ring homomorphisms and surjective group homomorphisms. We demonstrate that for finite cyclic structures, the number of ring homomorphisms from $\mathbb{Z}_m$ to $\mathbb{Z}_n$ is a divisor of the number of surjective group homomorphisms from $\mathbb{Z}_m$ to $\mathbb{Z}_n$, where $n$ is not of the form $2 \cdot \alpha$, where each prime factor $p$ of $\alpha$ satisfies $p \equiv 3 \pmod{4}$. We further extend this result for finite abelian structures.