A new characterization of $A_{p}$ with $p $ and $p-2$ are twin primes

Abstract

Let $G$ be a finite group and $\pi_{e}(G)$ be the set of elements order of $G$. Let $k \in \pi_{e}(G)$ and $m_{k}$ be the number of elements of order $k$ in $G$. Set nse($G$):=$\{ m_{k} | k \in \pi_{e}(G)\}$. Assume $p$ and $p-2$ are twin primes. We prove that if $G$ is a group such that nse($G$)=nse($A_{p}$) and $p\in \pi (G)$, then $G \cong A_{p}$. As a consequence of our results we prove that $A_{p}$ is uniquely determined by its nse and order.

Keywords

• Element order
• set of the numbers of elements of the same order
• alternating group