# 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 ${U}_{5}\left(2\right)$.

## How to cite

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Mohammadzadeh, A., and Moghaddamfar, Ali Reza. "Several quantitative characterizations of some specific groups." Commentationes Mathematicae Universitatis Carolinae 58.1 (2017): 19-34. <http://eudml.org/doc/287885>.

abstract = {Let $G$ be a finite group and let $\pi (G)=\lbrace p_1, p_2,\ldots , p_k\rbrace$ be the set of prime divisors of $|G|$ for which $p_1< p_2< \cdots < p_k$. The Gruenberg-Kegel graph of $G$, denoted $\operatorname\{GK\} (G)$, 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 $p_i p_j$. The degree of a vertex $p_i$ in $\{\rm GK\}(G)$ is denoted by $d_G(p_i)$ and the $k$-tuple $D(G)= (d_G(p_1), d_G(p_2),\ldots , d_G(p_k))$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi (G)$ is the vertex set of a connected component of $\operatorname\{GK\} (G)$, then the largest $\omega$-number which divides $|G|$, is said to be an order component of $\operatorname\{GK\} (G)$. 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 $U_5(2)$.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {OD-characterization of finite group; prime graph; degree pattern; simple group; $2$-Frobenius group},
language = {eng},
number = {1},
pages = {19-34},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Several quantitative characterizations of some specific groups},
url = {http://eudml.org/doc/287885},
volume = {58},
year = {2017},
}

TY - JOUR
TI - Several quantitative characterizations of some specific groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 1
SP - 19
EP - 34
AB - Let $G$ be a finite group and let $\pi (G)=\lbrace p_1, p_2,\ldots , p_k\rbrace$ be the set of prime divisors of $|G|$ for which $p_1< p_2< \cdots < p_k$. The Gruenberg-Kegel graph of $G$, denoted $\operatorname{GK} (G)$, 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 $p_i p_j$. The degree of a vertex $p_i$ in ${\rm GK}(G)$ is denoted by $d_G(p_i)$ and the $k$-tuple $D(G)= (d_G(p_1), d_G(p_2),\ldots , d_G(p_k))$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi (G)$ is the vertex set of a connected component of $\operatorname{GK} (G)$, then the largest $\omega$-number which divides $|G|$, is said to be an order component of $\operatorname{GK} (G)$. 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 $U_5(2)$.
LA - eng
KW - OD-characterization of finite group; prime graph; degree pattern; simple group; $2$-Frobenius group
UR - http://eudml.org/doc/287885
ER -

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