# The invariably generating graph of the alternating and symmetric groups

## TL;DR

This paper investigates the structure of the invariably generating graph of alternating and symmetric groups, showing that after removing isolated vertices, the graph remains connected with a diameter of at most 6.

## Contribution

It provides the first detailed analysis of the connectivity and diameter of the invariably generating graph for these groups, revealing its bounded diameter.

## Key findings

- The graph is connected after removing isolated vertices.
- The diameter of the graph is at most 6.
- The structure of the graph depends on the conjugacy classes of the groups.

## Abstract

Given a finite group $G$, the invariably generating graph of $G$ is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of $G$, and two classes are adjacent if and only if they invariably generate $G$. In this paper we study this object for alternating and symmetric groups. The main result of the paper states that, if we remove the isolated vertices from the graph, the resulting graph is connected and has diameter at most $6$.