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

TL;DR

The paper introduces uni-width subgroups and the uni-core $U(1;G)$, showing it is the largest uni-width subgroup and sits inside the Fitting subgroup ${\mathrm{Fit}}(G)$. It then characterizes when a finite group has a nontrivial universal element, proving a precise dichotomy: such an element exists iff the group is cyclic or a generalized quaternion $2$-group, with implications for the structure of $\Gamma_G$ and its central elements. A detailed analysis of cyclic classes and singular orders leads to conditions under which the $L(2,1)$-lambda number satisfies $\lambda(G)=|G|$, including a constructive Hamiltonian-path approach in the complement of the power graph. Finally, the authors present an infinite family where $\lambda(G) > |G|$, establishing the optimality of their criteria. Overall, the work links uni-width structure, power-graph topology, and vertex-coloring constraints in finite groups, with explicit results for both typical and exceptional group families.

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 $Γ_G$ 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 $Γ_G$ 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 $λ(G)$ 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 $Γ_G$, which is known to be $\geq |G|$. Generalizing an earlier result, we then derive a necessary condition on a finite group $G$ such that $λ(G) = |G|$. Finally, we show that this result is best possible by exhibiting a family of groups without the necessary condition for which $λ(G) > |G|$.

Figures (2)

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Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• Lemma 2.4
• Remark 2.5
• Lemma 3.1