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A characterization of \(A_{5}\) by its Same-order type

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Abstract

Let G be a group. The same-order type of G may be defined to be the set of sizes of equivalence classes for the equivalence relation \(\thicksim \) on G defined by

$$\begin{aligned} \forall \ g, h \in G \ \ g\thicksim h \Longleftrightarrow |g|=|h|. \end{aligned}$$

Shen et al. (Monatsh Math 160:337–341,2010), showed that \(A_5\) is the only group with the same-order type \(\{1,15,20,24\}\). In this paper, among other things, we prove that a nonabelian simple group G has same-order type with just four members if and only if \(G\cong A_5\).

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Acknowledgments

The authors would like to thank the referee for careful reading and helpful comments. This research was supported by the University of Kurdistan.

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

    L. Jafari Taghvasani & M. Zarrin

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  1. L. Jafari Taghvasani
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  2. M. Zarrin
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Correspondence to M. Zarrin.

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Communicated by A. Constantin.

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Taghvasani, L.J., Zarrin, M. A characterization of \(A_{5}\) by its Same-order type. Monatsh Math 182, 731–736 (2017). https://doi.org/10.1007/s00605-016-0950-9

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