Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24493120220201Characterization of some alternating groups by order and largest element order3544457110.22060/ajmc.2021.19507.1047ENAliMahmoudifarDepartment of Mathematics, North Tehran Branch, Islamic Azad University, Tehran, IranAyoubGharibkhajehDepartment of Mathematics, North Tehran Branch, Islamic Azad University, Tehran, IranJournal Article20210115The prime graph (or Gruenberg-Kegel graph) of a finite group is a well-known graph. In this paper, first, we investigate the structure of the finite groups with a non-complete prime graph. Then as an application, we prove that every alternating group $A_{n}$, where $nleq31$ is determined by its order and its largest element order. Also, we show that $A_{32}$ is not characterizable by order and the largest element order.https://ajmc.aut.ac.ir/article_4571_a2f64b4f0ce8012d4af43a92072ed287.pdf