A characterization of some projective special linear groups

TL;DR

This paper demonstrates that for certain small projective special linear groups, their order and prime graph degree patterns uniquely identify them among finite groups.

Contribution

It proves that $L_3(q)$ groups with $q<100$ are uniquely characterized by their order and prime graph degree patterns.

Findings

Groups $L_3(q)$ with $q<100$ are uniquely determined by their order and prime graph degree patterns.

Any finite group with the same order and prime graph degree pattern as $L_3(q)$ is isomorphic to it.

The result provides a new characterization method for these groups based on prime graph properties.

Abstract

In this paper, we show that projective special linear groups $S:=L_3(q)$ with $q$ less than $100$ are uniquely determined by their orders and degree patterns of their prime graphs. Indeed, we prove that if $G$ is a finite group whose order and degree pattern of its prime graph is the same as the order and the degree pattern of $S$, then $G$ is isomorphic to $S$.