Characterization of some alternating groups by order and the largest element order

TL;DR

This paper studies the structure of finite groups with non-complete prime graphs and proves that certain alternating groups are uniquely identified by their order and largest element order.

Contribution

It characterizes finite groups with non-complete prime graphs and establishes that specific alternating groups are uniquely determined by their order and largest element order.

Findings

Finite groups with non-complete prime graphs are analyzed.

Alternating groups A_n for n ≤ 20 or n = 23, 24 are uniquely identified by order and largest element order.

Abstract

The prime graph (or Gruenberg-Kegel graph) of a finite group $G$ is a familiar graph. In this paper first, we investigate the structure of the finite groups with a non-complete prime graph. Then we prove that every alternating group $A_{n}$, where $n\leq20$ or $n\in\{23,24\}$ is determined by its order and its largest element order.