# Several quantitative characterizations of some specific groups

• Volume: 58, Issue: 1, page 19-34
• ISSN: 0010-2628

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## Abstract

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Let $G$ be a finite group and let $\pi \left(G\right)=\left\{{p}_{1},{p}_{2},...,{p}_{k}\right\}$ be the set of prime divisors of $|G|$ for which ${p}_{1}<{p}_{2}<\cdots <{p}_{k}$. The Gruenberg-Kegel graph of $G$, denoted $GK\left(G\right)$, is defined as follows: its vertex set is $\pi \left(G\right)$ and two different vertices ${p}_{i}$ and ${p}_{j}$ are adjacent by an edge if and only if $G$ contains an element of order ${p}_{i}{p}_{j}$. The degree of a vertex ${p}_{i}$ in $\mathrm{GK}\left(G\right)$ is denoted by ${d}_{G}\left({p}_{i}\right)$ and the $k$-tuple $D\left(G\right)=\left({d}_{G}\left({p}_{1}\right),{d}_{G}\left({p}_{2}\right),...,{d}_{G}\left({p}_{k}\right)\right)$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi \left(G\right)$ is the vertex set of a connected component of $GK\left(G\right)$, then the largest $\omega$-number which divides $|G|$, is said to be an order component of $GK\left(G\right)$. 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 ${U}_{4}\left(2\right)$. Second, we prove that there are exactly two non-isomorphic finite groups with the same order components as ${U}_{5}\left(2\right)$.

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