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

The Gruenberg-Kegel graph ${\rm GK}(G)=(V_G, E_G)$ 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 ${\rm deg}_G(p)$ of a vertex $p\in V_G$ is the number of edges incident on $p$. In the case when $\pi(G)=\{p_1, p_2,..., p_h\}$ with $p_1< p_2< ... < p_h$, we consider the $h$-tuple $D(G)=({\rm deg}_G(p_1), {\rm deg}_G(p_2),..., {\rm deg}_G(p_h))$, which is called the degree pattern of $G$. The group $G$ is called $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $H$ satisfying condition $(|H|, D(H))=(|G|, D(G))$. 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 ${\rm Aut}(L_p(2))$ and ${\rm Aut}(L_{p+1}(2))$, where $2^p-1$ is a Mersenne prime, are OD-characterizable (Theorem \ref{auto}).