On the structure of $LC$-nilpotent groups

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.