An exact upper bound for sums of element orders in non-cyclic finite groups

By Marcel Herzog, P. Longobardi, M. Maj. Journal of Pure and Applied Algebra, 2016

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Abstract Denote the sum of element orders in a finite group G by ψ ( G ) and let C n denote the cyclic group of order n . Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n . We proved that ψ ( G ) ≤ 7 11 ψ ( C n ) and ψ ( G ) 1 q − 1 ψ ( C n ) . The first result is best possible, since for each n = 4 k , k odd, there exists a group G of order n satisfying ψ ( G ) = 7 11 ψ ( C n ) and the second result implies that if G is of odd order, then ψ ( G ) 1 2 ψ ( C n ) . Our results improve the inequality ψ ( G ) ψ ( C n ) obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some ψ ( G ) -based sufficient conditions for the solvability of G .

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/186f15527b3a4044eeb4115e2174c6fc1de533ec

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