Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Maryam Khatami; Azam Babai

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 1, page 121-130
  • ISSN: 0011-4642

Abstract

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Let G be a finite group. An element g ∈ G is called a vanishing element if there exists an irreducible complex character χ of G such that χ ( g ) = 0 . Denote by Vo ( G ) the set of orders of vanishing elements of G . Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo ( G ) = Vo ( M ) and | G | = | M | . Then G ≅ M . We answer in affirmative this conjecture for M = S z ( q ) , where q = 2 2 n + 1 and either q - 1 , q - 2 q + 1 or q + 2 q + 1 is a prime number, and M = F 4 ( q ) , where q = 2 n and either q 4 + 1 or q 4 - q 2 + 1 is a prime number.

How to cite

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Khatami, Maryam, and Babai, Azam. "Recognition of some families of finite simple groups by order and set of orders of vanishing elements." Czechoslovak Mathematical Journal 68.1 (2018): 121-130. <http://eudml.org/doc/294296>.

@article{Khatami2018,
abstract = {Let $G$ be a finite group. An element $g\in G$ is called a vanishing element if there exists an irreducible complex character $\chi $ of $G$ such that $\chi (g)=0$. Denote by $\{\rm Vo\}(G)$ the set of orders of vanishing elements of $G$. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let $G$ be a finite group and $M$ a finite nonabelian simple group such that $\{\rm Vo\}(G)=\{\rm Vo\}(M)$ and $|G|=|M|$. Then $G\cong M$. We answer in affirmative this conjecture for $M=Sz(q)$, where $q=2^\{2n+1\}$ and either $q-1$, $q-\sqrt\{2q\}+1$ or $q+\sqrt\{2q\}+1$ is a prime number, and $M=F_4(q)$, where $q=2^n$ and either $q^4+1$ or $q^4-q^2+1$ is a prime number.},
author = {Khatami, Maryam, Babai, Azam},
journal = {Czechoslovak Mathematical Journal},
keywords = {finite simple groups; vanishing element; vanishing prime graph},
language = {eng},
number = {1},
pages = {121-130},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Recognition of some families of finite simple groups by order and set of orders of vanishing elements},
url = {http://eudml.org/doc/294296},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Khatami, Maryam
AU - Babai, Azam
TI - Recognition of some families of finite simple groups by order and set of orders of vanishing elements
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 1
SP - 121
EP - 130
AB - Let $G$ be a finite group. An element $g\in G$ is called a vanishing element if there exists an irreducible complex character $\chi $ of $G$ such that $\chi (g)=0$. Denote by ${\rm Vo}(G)$ the set of orders of vanishing elements of $G$. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let $G$ be a finite group and $M$ a finite nonabelian simple group such that ${\rm Vo}(G)={\rm Vo}(M)$ and $|G|=|M|$. Then $G\cong M$. We answer in affirmative this conjecture for $M=Sz(q)$, where $q=2^{2n+1}$ and either $q-1$, $q-\sqrt{2q}+1$ or $q+\sqrt{2q}+1$ is a prime number, and $M=F_4(q)$, where $q=2^n$ and either $q^4+1$ or $q^4-q^2+1$ is a prime number.
LA - eng
KW - finite simple groups; vanishing element; vanishing prime graph
UR - http://eudml.org/doc/294296
ER -

References

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