# OD-characterization of almost simple groups related to $L_{2}(49)$

## Liangcai Zhang and Wujie Shi

Address:

Liangcai Zhang College of Mathematics and Physics, Chongqing University, Shapingba, Chongqing 400044, People’s Republic of China School of Mathematical Sciences, Suzhou University, Suzhou, Jiangsu 215006, People’s Republic of China

Wujie Shi School of Mathematical Sciences, Suzhou University, Suzhou, Jiangsu 215006, People’s Republic of China

E-mail:

zlc213@163.com

wjshi@suda.edu.cn

Abstract: In the present paper, we classify groups with the same order and degree pattern as an almost simple group related to the projective special linear simple group $L_{2}(49)$. As a consequence of this result we can give a positive answer to a conjecture of W. J. Shi and J. X. Bi, for all almost simple groups related to $L_{2}(49)$ except $L_{2}(49)\cdot 2^{2}$. Also, we prove that if $M$ is an almost simple group related to $L_{2}(49)$ except $L_{2}(49)\cdot 2^{2}$ and $G$ is a finite group such that $|G|=|M|$ and $\Gamma (G)=\Gamma (M)$, then $G\cong M$.

AMSclassification: primary 20D05; secondary 20D06, 20D60.

Keywords: almost simple group, prime graph, degree of a vertex, degree pattern.