Abstract
For a group G, write \(g \sim h\) if \(g, h \in G\) have the same order. The set of sizes of the equivalence classes with respect to this relation is called the same-order type of G; thus it is the set with numbers of elements of each order. In this article we prove that a group is isomorphic to the alternating group \(A_5\) if and only if the same-order type of G is \(\{1,pq,4p,8q\}\) with the p and q primes.
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Acknowledgments
The authors would like to thank Professor John Wilson for many helpful comments and suggestions that lead to improvement the original manuscript.
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Communicated by A. Constantin.
Project supported by NSF of China (No. 11201133, No. 11171364, No. 11271301) and the Innovation Foundation of Chongqing (KJTD201321).
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Shen, R., Zou, X. & Shi, W. A characterization of \(A_5\) by same-order type. Monatsh Math 182, 127–142 (2017). https://doi.org/10.1007/s00605-015-0845-1
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DOI: https://doi.org/10.1007/s00605-015-0845-1