Article

 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 . Category: math . 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 . 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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