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

TL;DR

This paper investigates the prime graph degree patterns of finite groups, establishes bounds on the sum of degrees, analyzes simple Lie type groups, and proves certain linear groups are uniquely characterized by their degree patterns.

Contribution

It provides bounds on the sum of degrees in prime graphs, details degree patterns for finite simple groups of Lie type, and proves specific linear groups are OD-characterizable.

Findings

Sharp bounds on the sum of degrees in prime graphs.

Degree patterns for finite simple groups of Lie type.

Linear groups L_4(19), L_4(23), L_4(27), L_4(29), L_4(31), L_4(32), L_4(37) are OD-characterizable.

Abstract

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).