A Strict Inequality for a Minimal Degree of a Direct Product

TL;DR

This paper presents a counter-example showing that the minimal faithful permutation degree of a direct product of finite groups can be strictly less than the sum of their individual minimal degrees, challenging previous assumptions.

Contribution

It provides the first known counter-example of degree 12, disproving the conjecture that the minimal degree of a direct product always equals the sum of individual degrees.

Findings

Counter-example of degree 12 for the inequality

Disproves Wright's conjecture for all finite groups

Shows minimal degree of direct product can be strictly less

Abstract

The minimal faithful permutation degree of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright established conditions for when the minimal degree of a direct product equals the sum of the minimal degrees for two finite groups. Wright asked whether this is true for all finite groups. A counter- example of degree 15 was provided by the referee and was added as an addendum in a paper of Wright. Here we provide a counter-example of degree 12.