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

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 $\lambda(G) > |G|$.