On some problems regarding $LCM$-groups

TL;DR

This paper studies finite groups through element-order divisibility, defining $LCM$-groups via $LCM(G)=G$ and related concepts such as $CP_2$ and $P_2^*$. It provides a structural classification of minimal non-$LCM$-groups, along with criteria for nilpotency and the $LCM$-property using the ratios $lcm(G)$ and $lcm^*(G)$. It also shows that minimum $\mathcal{F}$-covers of sets of $LCM$-groups need not be $LCM$, and answers Amiri's questions with extensive counterexamples, highlighting limits of order-based invariants in deducing group structure. Collectively, the results illuminate how element orders constrain group structure, subgroups, direct products, and covers in finite 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.