recognition by degree prime-power graph and order of some characteristically simple groups

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≤ 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 ≤ p are uniquely determined by their degree prime-power graphs and orders.