A characterization of A_5 by its Same-order type

TL;DR

This paper characterizes the alternating group A5 uniquely by its same-order type, showing that any nonabelian simple group with this same-order type is isomorphic to A5.

Contribution

It proves that a nonabelian simple group with the same-order type as A5 must be isomorphic to A5, providing a new characterization of A5 among simple groups.

Findings

A5 is uniquely characterized by its same-order type among nonabelian simple groups.

Nonabelian simple groups with the same-order type as A5 are isomorphic to A5.

The paper extends previous results on the same-order type of groups.

Abstract

Let G be a group, define an equivalence relation s as below: 8 g; h 2 G g s h () jgj = jhj the set of sizes of equivalence classes with respect to this relation is called the same-order type of G. Shen et al. (Monatsh. Math. 160 (2010), 337-341.), showed that A5 is the only group with the same-order type f1; 15; 20; 24g. In this paper, among other things, we prove that a nonabelian simple group G has same-order type fr; m; n; kg if and only if G ?= A5.