Document Type : Original Article

**Authors**

Department of Mathematics, North Tehran Branch, Islamic Azad University, Tehran, Iran

**Abstract**

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.

**Keywords**

**Main Subjects**

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February 2022

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