On the tree-number of the power graph associated with a finite groups

TL;DR

This paper investigates the structure of the power graph of finite groups, focusing on the number of spanning trees, and shows that the simple group A6 is uniquely identified by this tree-number among finite simple groups.

Contribution

It introduces properties of the power graph of finite groups, explores divisors of the tree-number, and proves the uniqueness of A6 based on its power graph's tree-number.

Findings

The power graph of any finite group is connected.

Divisors of the tree-number are characterized for certain groups.

A6 is uniquely identified by its power graph's tree-number among finite simple groups.

Abstract

Given a group $G$, we define the power graph $\mathcal{P}(G)$ as follows: the vertices are the elements of $G$ and two vertices $x$ and $y$ are joined by an edge if $\langle x \rangle \subseteq \langle y \rangle$ or $\langle y \rangle \subseteq \langle x \rangle$. Obviously the power graph of any group is always connected, because the identity element of the group is adjacent to all other vertices. We consider $\kappa(G)$, the number of spanning trees of the power graph associated with a finite group $G$. In this paper, for a finite group $G$, first we represent some properties of $\mathcal{P}(G)$, then we are going to find some divisors of $\kappa(G)$, and finally we prove that the simple group $A_6\cong L_2(9)$ is uniquely determined by tree-number of its power graph among all finite simple groups.