Another criterion for solvability of finite groups

Marcel Herzog; Patrizia Longobardi; Mercede Maj

doi:10.1016/j.jalgebra.2022.01.005

Another criterion for solvability of finite groups

Marcel Herzog, Patrizia Longobardi, Mercede Maj

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

Let G be a finite group. Denote by ψ(G) the sum ψ(G)=∑x∈G|x|, where |x| denotes the order of the element x, and by o(G) the quotient [Formula presented] Confirming a conjecture posed by E.I. Khukhro, A. Moreto and M. Zarrin, we prove that if o(G)<o(A₅), then G is solvable.

Original language	English
Pages (from-to)	1-23
Number of pages	23
Journal	Journal of Algebra
Volume	597
DOIs	https://doi.org/10.1016/j.jalgebra.2022.01.005
State	Published - 1 May 2022

Keywords

Average order
Element orders
Simple groups
Soluble groups

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10.1016/j.jalgebra.2022.01.005

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abstract = "Let G be a finite group. Denote by ψ(G) the sum ψ(G)=∑x∈G|x|, where |x| denotes the order of the element x, and by o(G) the quotient [Formula presented] Confirming a conjecture posed by E.I. Khukhro, A. Moreto and M. Zarrin, we prove that if o(G)5), then G is solvable.",

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N2 - Let G be a finite group. Denote by ψ(G) the sum ψ(G)=∑x∈G|x|, where |x| denotes the order of the element x, and by o(G) the quotient [Formula presented] Confirming a conjecture posed by E.I. Khukhro, A. Moreto and M. Zarrin, we prove that if o(G)5), then G is solvable.

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KW - Element orders

KW - Simple groups

KW - Soluble groups

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Another criterion for solvability of finite groups

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Keywords

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