# On the composition factors of a group with the same prime graph as ${B}_{n}\left(5\right)$

• Volume: 62, Issue: 2, page 469-486
• ISSN: 0011-4642

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## Abstract

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Let $G$ be a finite group. The prime graph of $G$ is a graph whose vertex set is the set of prime divisors of $|G|$ and two distinct primes $p$ and $q$ are joined by an edge, whenever $G$ contains an element of order $pq$. The prime graph of $G$ is denoted by $\Gamma \left(G\right)$. It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if $G$ is a finite group such that $\Gamma \left(G\right)=\Gamma \left({B}_{n}\left(5\right)\right)$, where $n\ge 6$, then $G$ has a unique nonabelian composition factor isomorphic to ${B}_{n}\left(5\right)$ or ${C}_{n}\left(5\right)$.

## How to cite

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Babai, Azam, and Khosravi, Behrooz. "On the composition factors of a group with the same prime graph as $B_{n}(5)$." Czechoslovak Mathematical Journal 62.2 (2012): 469-486. <http://eudml.org/doc/246706>.

@article{Babai2012,
abstract = {Let $G$ be a finite group. The prime graph of $G$ is a graph whose vertex set is the set of prime divisors of $|G|$ and two distinct primes $p$ and $q$ are joined by an edge, whenever $G$ contains an element of order $pq$. The prime graph of $G$ is denoted by $\Gamma (G)$. It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if $G$ is a finite group such that $\Gamma (G)=\Gamma (B_\{n\}(5))$, where $n\ge 6$, then $G$ has a unique nonabelian composition factor isomorphic to $B_\{n\}(5)$ or $C_\{n\}(5)$.},
author = {Babai, Azam, Khosravi, Behrooz},
journal = {Czechoslovak Mathematical Journal},
keywords = {prime graph; simple group; recognition; quasirecognition; prime graphs; finite simple groups; recognition; quasirecognition; sets of element orders},
language = {eng},
number = {2},
pages = {469-486},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the composition factors of a group with the same prime graph as $B_\{n\}(5)$},
url = {http://eudml.org/doc/246706},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Babai, Azam
AU - Khosravi, Behrooz
TI - On the composition factors of a group with the same prime graph as $B_{n}(5)$
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 2
SP - 469
EP - 486
AB - Let $G$ be a finite group. The prime graph of $G$ is a graph whose vertex set is the set of prime divisors of $|G|$ and two distinct primes $p$ and $q$ are joined by an edge, whenever $G$ contains an element of order $pq$. The prime graph of $G$ is denoted by $\Gamma (G)$. It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if $G$ is a finite group such that $\Gamma (G)=\Gamma (B_{n}(5))$, where $n\ge 6$, then $G$ has a unique nonabelian composition factor isomorphic to $B_{n}(5)$ or $C_{n}(5)$.
LA - eng
KW - prime graph; simple group; recognition; quasirecognition; prime graphs; finite simple groups; recognition; quasirecognition; sets of element orders
UR - http://eudml.org/doc/246706
ER -

## References

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