Absorption of Direct Factors With Respect to the Minimal Faithful Permutation Degree of a Finite Group

TL;DR

This paper investigates how the minimal faithful permutation degree of a finite group behaves under direct product absorption, revealing conditions under which this degree remains unchanged and characterizing the structure of groups involved.

Contribution

It establishes a characterization of groups H for which the minimal faithful permutation degree of G remains unchanged when taking direct products, linking this to embeddings in specific group classes.

Findings

If μ(G)=μ(G×H), then H embeds in a product of an abelian group of odd order, a generalized quaternion 2-group, and a free abelian group.

The minimal faithful permutation degree μ(G^n) stabilizes only when G is trivial.

The paper provides structural insights into the groups that preserve the permutation degree under direct product absorption.

Abstract

The minimal faithful permutation degree $\mu(G)$ of a finite group $G$ is the least nonnegative integer $n$ such that $G$ embeds in the symmetric group $\Sym(n)$. We prove that if $H$ is a group then $\mu(G)=\mu(G\times H)$ for some group $G$ then $H$ embeds in $A\times Q^k$ for some abelian group of odd order, some generalised quaternion $2$-group and some nonnegative integer $k$. As a consequence, $\mu(G^{n+1})=\mu(G^n)$ for some nonnegative integer $n$ if and only if $G$ is trivial.