The invariably generating graph of the alternating and symmetric 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 connected if and only if they invariably generate G. In this paper we study this object for alternating and symmetric groups. First we observe that in most cases it has isolated vertices. Then, we prove that if we take them out we obtain a connected graph. Finally, we bound the diameter of this new graph from above and from below -apart from trivial cases, it is between 3 and 6 -, and in about half of the cases we compute it exactly. Preliminary observationsThe vertices of our graph are conjugacy classes of alternating and symmetric groups. A conjugacy class of S n is faithfully represented by its cycle type. The same holds for a conjugacy class of A n , unless its cycle type is made of distinct odd parts, in which case there are two conjugacy classes with the same cycle type. Except for the latter case, we will consider the vertices of the graph as partitions of n; we fix now some language about this.Given G a subgroup of S n and given (a 1 , . . . , a t ) a partition of n, we will say that (a 1 , . . . , a t ) belongs to G, or that (a 1 , . . . , a t ) is contained in G, if G contains elements with cycle type (a 1 , . . . , a t ).