On the structure of $LC$-nilpotent groups

TL;DR

This paper investigates the structure of $LC$-nilpotent groups, characterizing when finite solvable groups admit an $LC$-nilpotent series and establishing that such groups form a variety.

Contribution

It provides a characterization of finite solvable groups with an $LC$-nilpotent series and proves that all $LC$-nilpotent groups form a variety.

Findings

Finite solvable groups admit an $LC$-nilpotent series iff they lack a specific $2$-Frobenius subgroup.

The class of all $LC$-nilpotent groups constitutes a variety.

The subgroup $LC(G)$ is a nilpotent characteristic subgroup of $G$.

Abstract

For a finite group $G$, let $LC(G)$ be the subgroup generated by elements $x$ such that, for all $y \in G$ and all integers $n$, the order of $x^n y$ divides the least common multiple of the orders of $x$ and $y$. This subgroup is a nilpotent characteristic subgroup of $G$. In this article, among other results, we show that a finite solvable group $G$ admits an $LC$-nilpotent series if and only if $G$ does not contain any $2$-Frobenius subgroup of type $(p, q, p)$. As a consequence of this theorem, we conclude that the algebraic system comprising all $LC$-nilpotent groups forms a variety.