On the structure of $LC$-nilpotent groups

TL;DR

This work investigates the structure of $LC$-nilpotent finite groups by introducing and analyzing the $LC$- and $LCM_p$-subgroups arising from order-compatibility conditions on products. The authors establish a key solvable-group criterion: a finite solvable $G$ has an $LC$-nilpotent series if and only if no section of $G$ contains a $2$-Frobenius subgroup of type $(p,q,p)$, and they show that the class of $LC$-nilpotent groups forms a variety. They further classify LC-nilpotent groups of maximal $LC$-class, proving that non-abelian groups of order $pq$ (with $p<q$) are precisely the ones with maximal $LC$-class, and they determine the constrained structure when $p$-groups have cyclic quotients, including dihedral, semidihedral, and quaternion-type families. Overall, the results connect local element-order constraints to global group structure, offering a robust framework for recognizing and constructing LC-nilpotent finite groups within the landscape of finite solvable and supersolvable groups.

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.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• proof