Characterization of some projective special linear groups in dimension four by their orders and degree patterns

Abstract

‎Let $G$ be a finite group‎. ‎The degree pattern of $G$ denoted by‎ ‎$D(G)$ is defined as follows‎: ‎If $pi(G)={p_{1},p_{2},...,p_{k}}$ such that‎ ‎$p_{1}<p_{2}<...<p_{k}$‎, ‎then $D(G):=(deg(p_{1}),deg(p_{2}),...,deg(p_{k}))$‎, ‎where $deg(p_{i})$‎ ‎for $1leq ileq k$‎, ‎are the degree of vertices $p_{i}$ in the‎ ‎prime graph of $G$‎. ‎In this article, we consider a finite group $G$‎ ‎under assumptions $|G|=|L_{4}(2^{n})|$ and $D(G)=D(L_{4}(2^{n}))$‎, ‎where $nin{5‎, ‎6‎, ‎7}$ and we prove that $Gcong L_{4}(2^{n})$.