Magma Proof of Strict Inequalities for Minimal Degrees of Finite Groups

TL;DR

This paper presents a Magma-based proof demonstrating that the minimal faithful permutation degree of a finite group can be strictly less than the sum of the degrees of its factors, highlighting a non-additive property.

Contribution

It provides the first computational proof showing the minimal degree for a product of groups can be strictly less than the sum of their individual minimal degrees.

Findings

Confirmed that 10 is the smallest degree with such a property.

Established existence of groups G and H with $mma(G imes H) < mma(G) + mma(H).

Demonstrated the effectiveness of Magma in proving properties of minimal degrees.

Abstract

The minimal faithful permutation degree of a finite group $G$, denote by $\mu(G)$ is the least non-negative integer $n$ such that $G$ embeds inside the symmetric group $\Sym(n)$. In this paper, we outline a Magma proof that 10 is the smallest degree for which there are groups $G$ and $H$ such that $\mu(G \times H) < \mu(G)+ \mu(H)$.