Shen's conjecture on groups with given same order type

Document Type : Research Paper

Authors

1 Department of Mathematics, University of Kurdistan, P.O. Box: 416 Sanandaj, Iran

2 University of Kurdistan

Abstract

‎‎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 $|\alpha{(H)}|\leq 2$‎.

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References

[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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History

  • Receive Date: 09 June 2015
  • Revise Date: 23 August 2015
  • Accept Date: 29 August 2015
  • Published Online: 01 March 2017

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