A new characterization of some characteristically simple groups

Document Type : Original Article


Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran


Let $G$ be a finite group and ${\rm cd}(G)$ be the set of irreducible complex character degrees of $G$. It was proved that some finite simple groups are uniquely determined by their orders and their degree graphs. Recently, in [Behravesh, et al., Recognition of Janko groups and some simple $K_4$-groups by the order and one irreducible character degree or character degree graph, Int. J. Group Theory, DOI: $10.22108/{\rm ijgt}.2019.113029.1502$.] new characterizations for some finite simple groups are given. Also, in [Qin, et al., Mathieu groups, and its degree prime-power graphs, Comm. Algebra, $2019$] the degree prime-power graph of a finite group is introduced and it is proved that the Mathieu groups are uniquely determined by order and degree prime-power graph. In this paper, we continue this work and we characterize some simple groups and some characteristically simple groups by their orders and some vertices of their degree prime-power graphs.


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