# The diameter of proper power graphs of alternating groups

## TL;DR

This paper investigates the diameter of the proper power graph of alternating groups, establishing bounds between 6 and 11 for large n, and describes short paths within these graphs.

## Contribution

It improves the bounds on the diameter of proper power graphs of alternating groups for large n and characterizes short paths in these graphs.

## Key findings

- Diameter of An is between 6 and 11 for n ≥ 51.
- Proper power graphs are connected for large n.
- Short paths in these graphs are characterized.

## Abstract

The power graph of finite group G is a simple graph whose vertex set is G and two distinct elements a and b are adjacent if and only if one of them is a power of the other. The proper power graph of G is a graph which is obtained by deleting the identity vertex (the identity element of G). In this paper, we improve the diameter bound of proper power graph of alternating group of degree n which the graph is connected. We show that the diameter of An is between 6 and 11, if the n at least 51. We also describe a number of short paths in these power graphs.