A new characterization of symmetric group by NSE

Azam Babai; Zeinab Akhlaghi

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 2, page 427-437
  • ISSN: 0011-4642

Abstract

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Let G be a group and ω ( G ) be the set of element orders of G . Let k ∈ ω ( G ) and m k ( G ) be the number of elements of order k in G . Let nse ( G ) = { m k ( G ) : k ∈ ω ( G ) } . Assume r is a prime number and let G be a group such that nse ( G ) = nse ( S r ) , where S r is the symmetric group of degree r . In this paper we prove that G ≅ S r , if r divides the order of G and r 2 does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

How to cite

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Babai, Azam, and Akhlaghi, Zeinab. "A new characterization of symmetric group by NSE." Czechoslovak Mathematical Journal 67.2 (2017): 427-437. <http://eudml.org/doc/288225>.

@article{Babai2017,
abstract = {Let $G$ be a group and $\omega (G)$ be the set of element orders of $G$. Let $k\in \omega (G)$ and $m_k(G)$ be the number of elements of order $k$ in $G$. Let nse$(G) = \lbrace m_k(G) \colon k \in \omega (G)\rbrace $. Assume $r$ is a prime number and let $G$ be a group such that nse$(G)=$ nse$(S_r)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.},
author = {Babai, Azam, Akhlaghi, Zeinab},
journal = {Czechoslovak Mathematical Journal},
keywords = {set of the numbers of elements of the same order; prime graph},
language = {eng},
number = {2},
pages = {427-437},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A new characterization of symmetric group by NSE},
url = {http://eudml.org/doc/288225},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Babai, Azam
AU - Akhlaghi, Zeinab
TI - A new characterization of symmetric group by NSE
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 427
EP - 437
AB - Let $G$ be a group and $\omega (G)$ be the set of element orders of $G$. Let $k\in \omega (G)$ and $m_k(G)$ be the number of elements of order $k$ in $G$. Let nse$(G) = \lbrace m_k(G) \colon k \in \omega (G)\rbrace $. Assume $r$ is a prime number and let $G$ be a group such that nse$(G)=$ nse$(S_r)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.
LA - eng
KW - set of the numbers of elements of the same order; prime graph
UR - http://eudml.org/doc/288225
ER -

References

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