## Abstract

A finite group *G* is called \(\psi \)-divisible if \(\psi (H)|\psi (G)\) for any subgroup *H* of *G*, where \(\psi (H)\) and \(\psi (G)\) are the sum of element orders of *H* and *G*, respectively. In this paper, we classify the finite groups whose subgroups are all \(\psi \)-divisible. Since the existence of \(\psi \)-divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the \(\psi \)-divisibility property of ZM-groups. In the end, we introduce the concept of \(\psi \)-normal divisible group, i.e., a group for which the \(\psi \)-divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many \(\psi \)-normal divisible groups which are neither simple nor nilpotent.

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## Acknowledgements

The author is grateful to the reviewers for their remarks which improve the previous version of the paper.

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The author was supported by the European Social Fund, through Operational Programme Human Capital 2014-2020, Project No. POCU/380/6/13/123623.

Communicated by Kar Ping Shum.

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Lazorec, MS. On a Divisibility Property Involving the Sum of Element Orders.
*Bull. Malays. Math. Sci. Soc.* **44, **941–951 (2021). https://doi.org/10.1007/s40840-020-00987-8

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### Keywords

- Group element orders
- Sum of element orders
- ZM-groups

### Mathematics Subject Classification

- Primary 20D60
- Secondary 20D15
- 20D20