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

Abstract

In this paper, by the order of a group and triviality of $O_p(G)$ for some prime $p$, we give a new characterization for some characteristically simple groups. In fact, we prove that if {$p \in \{5, 17, 23, 37, 47, 73\}$ and $n \leqslant p$, where $n$ is a natural number, then $G\cong{{\rm PSL}(2,p)}^{n}$ if and only if $ |G|=|{{\rm PSL}(2,p)}|^{n}$ and $O_p(G)=1$. Recently in [Qin, Yan, Shum and Chen, Comm. Algebra, 2019], the degree prime-power graph of a finite groupb 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 ${\rm PSL}(2,p)^n$, where $p\in\{5,17,23,37,47,73\}$ and $n\leqslant{p}$ are uniquely determined by their degree prime-power graphs and orders.

Keywords

• degree prime power graph
• order
• characteristically simple group
• characterization