The Smallest Faithful Permutation Degree for a Direct Product Obeying an Inequality Condition

TL;DR

This paper investigates the minimal faithful permutation degree of direct product groups, establishing that for degrees up to 9, the degree of the product equals the sum of individual degrees, with 10 being the minimal counterexample.

Contribution

The paper proves that the minimal faithful permutation degree of a direct product of groups equals the sum of degrees for all cases up to 9, identifying 10 as the minimal degree where strict inequality can occur.

Findings

For degrees ≤ 9, μ(G×H) = μ(G) + μ(H).

10 is the smallest degree where μ(G×H) < μ(G) + μ(H).

The result characterizes the minimal faithful permutation degrees for small groups.

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 $\Sym(n)$. Clearly $\mu(G \times H) \le \mu(G) + \mu(H)$ for all finite groups $G$ and $H$. Wright (1975) proves that equality occurs when $G$ and $H$ are nilpotent and exhibits an example of strict inequality where $G\times H$ embeds in $\Sym(15)$. Saunders (2010) produces an infinite family of examples of permutation groups $G$ and $H$ where $\mu(G \times H) < \mu(G) + \mu(H)$, including the example of Wright's as a special case. The smallest groups in Saunders' class embed in $\Sym(10)$. In this paper we prove that 10 is minimal in the sense that $\mu(G \times H) = \mu(G) + \mu(H)$ for all groups $G$ and $H$ such that $\mu(G\times H)\le 9$.