Minimal groups of given representation dimension

TL;DR

This paper classifies finite groups where every proper subgroup has a strictly smaller minimal faithful representation dimension, focusing on abelian groups and those with dimension at most 3.

Contribution

It provides a classification of groups with minimal faithful representation dimension strictly greater than that of all proper subgroups, especially for abelian groups and low dimensions.

Findings

Classified such groups when G is abelian.

Classified such groups when rdim(G) ≤ 3.

Established properties of minimal faithful representations.

Abstract

For a finite group $G$, let $\text{rdim}(G)$ denote the smallest dimension of a faithful, complex linear representation of $G$. It is clear that $\text{rdim}(H)\leq \text{rdim}(G)$ for any subgroup $H$ of $G$. We consider $G$ with the property that $\text{rdim}(H)<\text{rdim}(G)$ whenever $H$ is a proper subgroup of $G$, in particular proving a classification of such groups when $G$ is abelian or $\text{rdim}(G)\leq 3$.