Characterization of some alternating groups by order and the 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.