Abstract
Given a finite group G, we denote by \(\psi (G)\) the sum of the element orders in G. In this article, we prove that if t is the number of nonidentity conjugacy classes in G, then \(\psi (G)=1+t|G|\) if and only if G is either a group of prime order or a nonabelian group of the square-free order with two prime divisors. Also we find a unique group with the second maximum sum of the element orders among all finite groups of the same square-free order.
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The authors would like to thank the referees for their careful reading and valuable comments.
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Communicated by Ang Miin Huey.
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Jafarian Amiri, S.M., Amiri, M. Sum of the Element Orders in Groups of the Square-Free Orders. Bull. Malays. Math. Sci. Soc. 40, 1025–1034 (2017). https://doi.org/10.1007/s40840-016-0353-z
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DOI: https://doi.org/10.1007/s40840-016-0353-z