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Keywords:
character degree graph; irreducible character; characteristically simple group; complex group algebra
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.
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