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Sum of the Element Orders in Groups of the Square-Free Orders

Sum of the Element Orders in Groups of the Square-Free Orders Given a finite group G, we denote by $$\psi (G)$$ ψ ( 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 $$\psi (G)=1+t|G|$$ ψ ( 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

Sum of the Element Orders in Groups of the Square-Free Orders

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References (7)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia
Subject
Mathematics; Mathematics, general; Applications of Mathematics
ISSN
0126-6705
eISSN
2180-4206
DOI
10.1007/s40840-016-0353-z
Publisher site
See Article on Publisher Site

Abstract

Given a finite group G, we denote by $$\psi (G)$$ ψ ( 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 $$\psi (G)=1+t|G|$$ ψ ( 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.

Journal

Bulletin of the Malaysian Mathematical Sciences SocietySpringer Journals

Published: May 2, 2016

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