Several Quantitative Characterizations of Some Specific Groups

TL;DR

This paper investigates the structure of finite groups through the Gruenberg-Kegel graph, solving the OD-characterization problem for certain simple groups and identifying the number of groups sharing specific graph-based properties.

Contribution

It completely solves the OD-characterization problem for all finite non-abelian simple groups with prime divisors up to 29 and determines the exact count of groups with identical order components for specific groups.

Findings

Exactly two non-isomorphic groups share the order and degree pattern with U_4(2).

Exactly two non-isomorphic groups share the same order components as U_5(2).

The paper provides a complete characterization for these classes of groups.

Abstract

Let $G$ be a finite group and let $\pi(G)=\{p_1, p_2, \ldots, p_k\}$ be the set of prime divisors of $|G|$ for which $p_1<p_2<\cdots<p_k$. The Gruenberg-Kegel graph of $G$, denoted ${\rm GK}(G)$, is defined as follows: its vertex set is $\pi(G)$ and two different vertices $p_i$ and $p_j$ are adjacent by an edge if and only if $G$ contains an element of order $p_ip_j$. The degree of a vertex $p_i$ in ${\rm GK}(G)$ is denoted by $d_G(p_i)$ and the $k$-tuple $D(G)=\left(d_G(p_1), d_G(p_2), \ldots, d_G(p_k)\right)$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi(G)$ is the vertex set of a connected component of ${\rm GK}(G)$, then the largest $\omega$-number which divides $|G|$, is said to be an order component of ${\rm GK}(G)$. We will say that the problem of OD-characterization is solved for a finite group if we find the number of pairwise non-isomorphic finite groups with the same order and degree pattern as the group under study. The purpose of this article is twofold. First, we completely solve the problem of OD-characterization for every finite non-abelian simple group with orders having prime divisors at most 29. In particular, we show that there are exactly two non-isomorphic finite groups with the same order and degree pattern as $U_4(2)$. Second, we prove that there are exactly two non-isomorphic finite groups with the same order components as $U_5(2)$.