On some graphs associated with the finite alternating groups

TL;DR

This paper investigates the structure and connectivity of various power-related graphs associated with finite alternating groups, providing formulas for the number of components and conditions for 2-connectivity.

Contribution

It determines the number of components and connectivity properties of proper power graphs and related graphs for alternating groups, establishing new criteria for 2-connectivity.

Findings

Power graph $P(A_n)$ is 2-connected iff $P( ext{type})$ is 2-connected under specific conditions.

Number of components of the graphs is explicitly determined.

Connectivity properties depend on prime-related conditions of $n$, $n-1$, $n-2$, etc.

Abstract

Let $P_0(A_n), \widetilde{P}_0(A_n), P_0(\mathcal{T}(A_n))$ and $\mathcal{O}_0(A_n)$ be respectively the proper power graph, the proper quotient power graph, the proper power type graph and the proper order graph of the alternating group $A_n$, for $n\geq 3.$ We determine the number of the components of those graphs. In particular, we prove that the power graph $P(A_n)$ is $2$-connected if and only if the power type graph $P(\mathcal{T}(A_n))$ is $2$-connected, if and only if either $n = 3$ or none of $n, n-1, n-2, \frac{n}{2}$ and ${\frac{n-1}{2}}$ is a prime. We also give some information on the properties of those components.