Document Type : Original Article

**Authors**

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

**Abstract**

In this paper, by the order of a group and triviality of Op(G) for some prime p, we give a new characterization for some characteristically simple groups. In fact, we prove that if p ∈ {5, 17, 23, 37, 47, 73} and n 6 p, where n is a natural number, then G ∼= PSL(2, p) ^{n} if and only if |G| = |PSL(2, p)|^{ n} and Op(G) = 1. Recently in [Qin, Yan, Shum and Chen, Comm. Algebra, 2019], the degree primepower graph of a finite group have been introduced and it is proved that the Mathieu groups are uniquely determined by their degree prime-power graphs and orders. As a consequence of our results, we show that PSL(2, p) ^{n}, where p ∈ {5, 17, 23, 37, 47, 73} and n 6 p are uniquely determined by their degree prime-power graphs and orders.

**Keywords**

**Main Subjects**

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February 2021

Pages 11-15