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Jafari Taghvasani, Leyli, Zarrin, Mohammad. (1395). Shen's conjecture on groups with given same order type. , 6(1), 17-20. doi: 10.22108/ijgt.2017.10631
Leyli Jafari Taghvasani; Mohammad Zarrin. "Shen's conjecture on groups with given same order type". , 6, 1, 1395, 17-20. doi: 10.22108/ijgt.2017.10631
Jafari Taghvasani, Leyli, Zarrin, Mohammad. (1395). 'Shen's conjecture on groups with given same order type', , 6(1), pp. 17-20. doi: 10.22108/ijgt.2017.10631
Jafari Taghvasani, Leyli, Zarrin, Mohammad. Shen's conjecture on groups with given same order type. , 1395; 6(1): 17-20. doi: 10.22108/ijgt.2017.10631

Shen's conjecture on groups with given same order type

International Journal of Group Theory
مقاله 41، دوره 6، شماره 1، خرداد 2017، صفحه 17-20    اصل مقاله (172.7 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2017.10631
نویسندگان
Leyli Jafari Taghvasani1؛ Mohammad Zarrin* 2
1Department of Mathematics, University of Kurdistan, P.O. Box: 416 Sanandaj, Iran
2University of Kurdistan
چکیده
‎‎For any group $G$‎, ‎we define an equivalence relation $\thicksim$ as below‎: ‎\[\forall \ g‎, ‎h \in G \ \ g\thicksim h \Longleftrightarrow |g|=|h|\]‎ ‎the set of sizes of equivalence classes with respect to this relation is called the same-order type of $G$ and denote by $\alpha{(G)}$‎. ‎In this paper‎, ‎we give a partial answer to a conjecture raised by Shen‎. ‎In fact‎, ‎we show that if $G$ is a nilpotent group‎, ‎then $|\pi(G)|\leq |\alpha{(G)}|$‎, ‎where $\pi(G)$ is the set of prime divisors of order of $G$‎. ‎Also we investigate the groups all of whose proper subgroups‎, ‎say $H$ have $|\alpha{(H)}|\leq 2$‎.
کلیدواژه‌ها
‎Nilpotent groups‎؛ ‎Same-order type‎؛ ‎Schmidt group
مراجع
[1] J. S. Williams, Prime graph comp onents of nite groups, J. Algebra, 69 (1981) 487-513.

[2] O. Yu. Schmidt, Groups all of whose subgroups are nilp otent, Mat. Sbornik, 31 (1924) 366-372.

[3] R. Shen, On groups with given same order typ es, Comm. Algebra, 40 (2012) 2140-2150.

[4] R. Shen, C. Shao, Q. Jiang, W. Shi and V. D. Mazurov, A new characterization A5 , Monatsh. Math., 160 (2010) 337-341.

[5] M. Zarrin, A generalization of Schmidt's Theorem on groups with all subgroups nilp otent, Arch. Math. (Basel), 99 (2012) 201-206.

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