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

Marcel Herzog, Patrizia Longobardi*, Mercede Maj

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

39 Scopus citations

Abstract

Denote the sum of element orders in a finite group G by ψ(G) and let Cn 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)≤[Formula presented]ψ(Cn) and ψ(G)<[Formula presented]ψ(Cn). The first result is best possible, since for each n=4k, k odd, there exists a group G of order n satisfying ψ(G)=[Formula presented]ψ(Cn) and the second result implies that if G is of odd order, then ψ(G)<[Formula presented]ψ(Cn). Our results improve the inequality ψ(G)<ψ(Cn) 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.

Original languageEnglish
Pages (from-to)1628-1642
Number of pages15
JournalJournal of Pure and Applied Algebra
Volume222
Issue number7
DOIs
StatePublished - Jul 2018

Funding

FundersFunder number
Istituto Nazionale di Alta Matematica "Francesco Severi"
Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni

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