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 $n\le 20$ or $n\in \{23,24\}$ is determined by its order and its largest element order.