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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