The invariably generating graph of the alternating and symmetric groups

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.

The minimal polynomials of powers of cycles in the ordinary representations of symmetric and alternating groups

Denote the alternating and symmetric groups of degree [Formula: see text] by [Formula: see text] and [Formula: see text], respectively. Consider a permutation [Formula: see text], all of whose nontrivial cycles are of the same length. We find the minimal polynomials of [Formula: see text] in the ordinary irreducible representations of [Formula: see text] and [Formula: see text].

Boolean lattices in finite alternating and symmetric groups

Given a group G and a subgroup H, we let $\mathcal {O}_G(H)$ denote the lattice of subgroups of G containing H. This article provides a classification of the subgroups H of G such that $\mathcal {O}_{G}(H)$ is Boolean of rank at least $3$ when G is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilisers of chains of regular partitions, and the other arises by taking stabilisers of chains of regular product structures. As an application, we prove in this case a conjecture on Boolean overgroup lattices related to the dual Ore’s theorem and to a problem of Kenneth Brown.