OD-Characterization of Some Linear Groups Over Binary Field and Their Automorphism

The Gruenberg-Kegel graph $\mathrm{G}\mathrm{K}(G)=(V_G,E_G)$ of a finite group $G$ is a simple graph with vertex set $V_G=\pi (G)$, the set of all primes dividing the order of $G$, and such that two distinct vertices $p$ and $q$ are joined by an edge, $\{p,q\}\in E_G$, if $G$ contains an element of order $pq$. The degree ${\mathrm{d}\mathrm{e}\mathrm{g}}_G(p)$ of a vertex $p\in V_G$ is the number of edges incident on $p$. In the case when $\pi (G)=\{p_1,p_2,\mathrm{.}\mathrm{.}\mathrm{.},p_h\}$ with $p_1<p_2<\mathrm{.}\mathrm{.}\mathrm{.}<p_h$, we consider the $h$-tuple $D(G)=({\rm deg}_G(p_1), {\rm deg}_G(p_2),…, {\rm deg}G(p_h))$,whichiscalledthedegreepatternof$G$\mathrm{.}Thegroup$G$ is called $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $H$ satisfying condition $(\mathrm{\mid }H\mathrm{\mid },D(H))=(\mathrm{\mid }G\mathrm{\mid },D(G))$. Especially, a 1-fold OD-characterizable group is simply called OD-characterizable. In this paper, we first find the degree pattern of the projevtive special linear groups over binary field $L_n(2)$ and among other results we prove that the simple groups $L_{10}(2)$ and $L_{11}(2)$ are OD-characterizable (Theorem \ref{10-11}). It is also shown that automorphism groups $\mathrm{A}\mathrm{u}\mathrm{t}(L_p(2))$ and ${\rm Aut}(L{p+1}(2))$,where$2^p-1$ is a Mersenne prime, are OD-characterizable (Theorem \ref{auto}).