On some graphs associated with the finite alternating groups

Daniela Bubboloni; Mohammadali Iranmanesh; Seyed Mostafa Shaker

paper

On some graphs associated with the finite alternating groups

arXiv:1412.7324

Abstract

Let $P_{0} (A_{n}), P_{0} (A_{n}), P_{0} (T (A_{n}))$ and $O_{0} (A_{n})$ 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 $P (T (A_{n}))$ 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.