# A new characterization of Mathieu groups

## Changguo Shao and Qinhui Jiang

Address: Department of Mathematics, Shangqiu Normal College Shangqiu 476000, People’s Republic of China

E-mail:

shaoguozi@163.com

syjqh2001@163.com

Abstract: Let $G$ be a finite group and $\operatorname{nse}(G)$ the set of numbers of elements with the same order in $G$. In this paper, we prove that a finite group $G$ is isomorphic to $M$, where $M$ is one of the Mathieu groups, if and only if the following hold:
(1) $|G|=|M|$,
(2) $\operatorname{nse}(G)=\operatorname{nse}(M)$.

AMSclassification: primary 20D60; secondary 20D06.

Keywords: finite group, solvable group, order of element.