Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 3 1 2022 02 01 Characterization of some alternating groups by order and largest element order 35 44 4571 10.22060/ajmc.2021.19507.1047 EN Ali Mahmoudifar Department of Mathematics, North Tehran Branch, Islamic Azad University, Tehran, Iran Ayoub Gharibkhajeh Department of Mathematics, North Tehran Branch, Islamic Azad University, Tehran, Iran Journal Article 2021 01 15 The 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 $n\leq 31$ 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_bfd2308e9e75263970f8079115edebbd.pdf