Recognition of characteristically simple group A 5 × A 5 by character degree graph and order

Maryam Khademi; Behrooz Khosravi

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 4, page 1149-1157
  • ISSN: 0011-4642

Abstract

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The character degree graph of a finite group G is the graph whose vertices are the prime divisors of the irreducible character degrees of G and two vertices p and q are joined by an edge if p q divides some irreducible character degree of G . It is proved that some simple groups are uniquely determined by their orders and their character degree graphs. But since the character degree graphs of the characteristically simple groups are complete, there are very narrow class of characteristically simple groups which are characterizable by this method. We prove that the characteristically simple group A 5 × A 5 is uniquely determined by its order and its character degree graph. We note that this is the first example of a non simple group which is determined by order and character degree graph. As a consequence of our result we conclude that A 5 × A 5 is uniquely determined by its complex group algebra.

How to cite

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Khademi, Maryam, and Khosravi, Behrooz. "Recognition of characteristically simple group $A_5\times A_5$ by character degree graph and order." Czechoslovak Mathematical Journal 68.4 (2018): 1149-1157. <http://eudml.org/doc/294584>.

@article{Khademi2018,
abstract = {The character degree graph of a finite group $G$ is the graph whose vertices are the prime divisors of the irreducible character degrees of $G$ and two vertices $p$ and $q$ are joined by an edge if $pq$ divides some irreducible character degree of $G$. It is proved that some simple groups are uniquely determined by their orders and their character degree graphs. But since the character degree graphs of the characteristically simple groups are complete, there are very narrow class of characteristically simple groups which are characterizable by this method. We prove that the characteristically simple group $A_5 \times A_5 $ is uniquely determined by its order and its character degree graph. We note that this is the first example of a non simple group which is determined by order and character degree graph. As a consequence of our result we conclude that $A_5\times A_5$ is uniquely determined by its complex group algebra.},
author = {Khademi, Maryam, Khosravi, Behrooz},
journal = {Czechoslovak Mathematical Journal},
keywords = {character degree graph; irreducible character; characteristically simple group; complex group algebra},
language = {eng},
number = {4},
pages = {1149-1157},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Recognition of characteristically simple group $A_5\times A_5$ by character degree graph and order},
url = {http://eudml.org/doc/294584},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Khademi, Maryam
AU - Khosravi, Behrooz
TI - Recognition of characteristically simple group $A_5\times A_5$ by character degree graph and order
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 4
SP - 1149
EP - 1157
AB - The character degree graph of a finite group $G$ is the graph whose vertices are the prime divisors of the irreducible character degrees of $G$ and two vertices $p$ and $q$ are joined by an edge if $pq$ divides some irreducible character degree of $G$. It is proved that some simple groups are uniquely determined by their orders and their character degree graphs. But since the character degree graphs of the characteristically simple groups are complete, there are very narrow class of characteristically simple groups which are characterizable by this method. We prove that the characteristically simple group $A_5 \times A_5 $ is uniquely determined by its order and its character degree graph. We note that this is the first example of a non simple group which is determined by order and character degree graph. As a consequence of our result we conclude that $A_5\times A_5$ is uniquely determined by its complex group algebra.
LA - eng
KW - character degree graph; irreducible character; characteristically simple group; complex group algebra
UR - http://eudml.org/doc/294584
ER -

References

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