Uni-width subgroups, universal elements, and lambda number of finite groups

TL;DR

This paper investigates special subgroup structures, universal elements in power graphs, and the lambda number in finite groups, revealing structural properties and coloring bounds that extend previous results.

Contribution

It introduces the concept of the largest uni-width subgroup, characterizes universal elements in power graphs, and establishes bounds on the lambda number for finite groups.

Findings

Existence of a unique largest uni-width subgroup in finite groups.

Power graph of a finite group has a non-identity universal element iff the group is cyclic or a generalized quaternion 2-group.

Necessary conditions for the lambda number to equal the group order are identified, with examples showing the bounds are tight.

Abstract

A cyclic subgroup $N$ of a finite group $G$ is called a uni-width subgroup of $G$ if $N$ is the unique cyclic subgroup of $G$ of order $|N|$. In this article, we prove that a finite group $G$ admits a unique largest uni-width subgroup denoted by $U(1;G)$. We then show that the prime factors of the order of $U(1;G)$ influence the structure decomposition of its Fitting subgroup ${\mathrm{Fit}}(G)$. A power graph $\Gamma_G$ of a finite group is defined by $G$ being its set of vertices, and a pair of distinct elements $x,y \in G$ are connected by an edge if either $x \in \langle y \rangle$ or $y \in \langle x \rangle$. A universal element of a graph is a vertex that is adjacent to each of the remaining vertices. Our following result shows that a power graph $\Gamma_G$ of a finite non-trivial group admits a non-identity universal element if and only if it is either cyclic or a generalized quaternion $2$-group. The lambda number $\lambda(G)$ of a finite group $G$ is a measure of the least number of colors required for an $L(2,1)$-type of vertex coloring on $\Gamma_G$, which is known to be $\geq |G|$. Generalizing an earlier result, we then derive a necessary condition on a finite group $G$ such that $\lambda(G) = |G|$. Finally, we show that this result is best possible by exhibiting a family of groups without the necessary condition for which $\lambda(G) > |G|$.