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Recognition by degree prime-power graph and order of some characteristically simple groups | ||
| AUT Journal of Mathematics and Computing | ||
| مقاله 2، دوره 2، شماره 1، اردیبهشت 2021، صفحه 11-15 اصل مقاله (324.6 K) | ||
| نوع مقاله: Original Article | ||
| شناسه دیجیتال (DOI): 10.22060/ajmc.2020.18418.1033 | ||
| نویسندگان | ||
| Afsane Bahri؛ Behrooz Khosravi* ؛ Morteza Baniasad Azad | ||
| Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Degree prime power graph؛ Order؛ Characteristically simple group؛ Characterization | ||
| مراجع | ||
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[1] M. Baniasad Azad, B. Khosravi, Recognition of some characteristically simple groups by their complex group algebra, Math. Rep (to appear).
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[9] M. B. Nathanson, Elementary Methods in Number Theory, Springer edition, NewYork, 2000.
[10] C. Qin, Y. Yan, K. P. Shum, G. Y. Chen, Mathieu groups and its degree prime-power graphs, Commun. Algebra, 47 (10) (2019) 4173-4180.
[11] H. Xu, G. Y. Chen, Y. Yan, A new characterization of simple K3-groups by their orders and large degrees of their irreducible characters, Commun. Algebra, 42 (2014) 5374-5380.
[12] A. V. Zavarnitsine, Finite simple groups with narrow prime spectrum, Sib. Elektron. Mat. Izv 6 (2009) 1-12. | ||
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