On some problems regarding $LCM$-groups

TL;DR

This paper investigates the properties and structure of finite groups called $LCM$-groups, providing new criteria for their classification, analyzing minimal non-$LCM$-groups, and addressing open questions about their covers.

Contribution

It extends existing results on $LCM$-groups by characterizing their structure, establishing criteria for $LCM$-group classification, and exploring properties of their covers.

Findings

Characterization of minimal non-$LCM$-groups

Criteria for a group to be an $LCM$-group or nilpotent

Minimum covers of $LCM$-groups are generally not $LCM$-groups

Abstract

Let $G$ be a finite group and denote by $o(g)$ the order of an element $g\in G$. We say that $G$ is an $LCM$-group if $o(x^ny)$ is a divisor of the least common multiple of $o(x^n)$ and $o(y)$ for all $x, y\in G$ and $n\in\mathbb{N}$. This paper extends some existing results on $LCM$-groups, such as the structure of a minimal non-$LCM$-group, and establishes criteria for $G$ to be an $LCM$-group or a nilpotent group. We also prove that, in general, a minimum cover of a finite set of $LCM$-groups is not an $LCM$-group, and we answer two questions posed by M. Amiri.