OD-Characterization of Certain Four Dimensional Linear Groups with Related Results Concerning Degree Patterns

The prime graph of a finite group $G$, which is denoted by ${\rm GK}(G)$, is a simple graph whose vertex set is comprised of the prime divisors of $|G|$ and two distinct prime divisors $p$ and $q$ are joined by an edge if and only if there exists an element of order $pq$ in $G$. Let $p_1<p_2<...<p_k$ be all prime divisors of $|G|$. Then the degree pattern of $G$ is defined as ${\rm D}(G)=(deg_G(p_1), deg_G(p_2),..., deg_G(p_k))$, where $deg_G(p)$ signifies the degree of the vertex $p$ in ${\rm GK}(G)$. A finite group $H$ is said to be OD-characterizable if $G\cong H$ for every finite group $G$ such that $|G|=|H|$ and ${\rm D}(G)={\rm D}(H)$. The purpose of this article is threefold. First, it finds sharp upper and lower bounds on $\vartheta(G)$, the sum of degrees of all vertices in ${\rm GK}(G)$, for any finite group $G$ (Theorem 2.1). Second, it provides the degree of vertices 2 and the characteristic $p$ of the base field of any finite simple group of Lie type in their prime graphs (Propositions 3.1-3.7). Third, it proves the linear groups $L_4(19)$, $L_4(23)$, $L_4(27)$, $L_4(29)$, $L_4(31)$, $L_4(32)$ and $L_4(37)$ are OD-characterizable (Theorem 4.2).