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Title: Recognition of characteristically simple group $A_5\times A_5$ by character degree graph and order (English)
Author: Khademi, Maryam
Author: Khosravi, Behrooz
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 68
Issue: 4
Year: 2018
Pages: 1149-1157
Summary lang: English
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Category: math
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Summary: 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. \endgraf 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. (English)
Keyword: character degree graph
Keyword: irreducible character
Keyword: characteristically simple group
Keyword: complex group algebra
MSC: 20C15
MSC: 20D05
MSC: 20D08
MSC: 20D60
idZBL: Zbl 07031705
idMR: MR3881904
DOI: 10.21136/CMJ.2018.0134-17
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Date available: 2018-12-07T06:25:58Z
Last updated: 2019-05-16
Stable URL: http://hdl.handle.net/10338.dmlcz/147529
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