A criterion for nilpotency of a finite group by the sum of element orders

2019/03/23 by Tărnăuceanu, Marius
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1903.09744

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. In this paper, we prove that if |G|=n and ψ(G)>(13)/(21) ψ(C_n), then G is nilpotent. Moreover, we have ψ(G)=(13)/(21) ψ(C_n) if and only if n=6m with (6,m)=1 and G≅ S₃× C_m. Two interesting consequences of this result are also presented.

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