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This work aims to establish a criterion for the solvability of finite groups based on the harmonic mean of their element orders.

July 4, 2026Beiträge zur Algebra und Geometrie / Contributions to Algebra and GeometryOpen Access

A criterion for solvability of a finite group by the harmonic mean of element orders

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This work aims to establish a criterion for the solvability of finite groups based on the harmonic mean of their element orders.
Proved the relationship between the harmonic mean of element orders and group solvability.
Analyzed the implications of the bound on harmonic mean for finite groups.
Demonstrated that if h_m(G) < 60/23, then G is solvable.

Implication

Short note proves that groups with harmonic mean of element orders below a threshold are solvable.

Abstract

Let G be a finite group and hₘ(G) h m ( G ) be the harmonic mean of element orders of G . In this short note, we prove that if hₘ(G)<60/23 h m ( G ) < 60 23 , then G is solvable.

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Pleşcă et al. (2026) studied this question.

synapsesocial.com/papers/6a48a72989561a0c2d78eb98 https://doi.org/10.1007/s13366-026-00862-w

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Implication

Short note proves that groups with harmonic mean of element orders below a threshold are solvable.

Abstract

Let G be a finite group and hₘ(G) h m ( G ) be the harmonic mean of element orders of G . In this short note, we prove that if hₘ(G)<60/23 h m ( G ) < 60 23 , then G is solvable.

A criterion for solvability of a finite group by the harmonic mean of element orders

Key Points

Implication

Abstract

Discussion

Cite This Study

A criterion for solvability of a finite group by the harmonic mean of element orders

Key Points

Implication

Abstract

Discussion

Cite This Study

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