Irredundant generating sets and dimension-like invariants of the finite group

TL;DR

This paper extends the classification of irredundant generating sets in symmetric and alternating groups, and derives formulas for dimension-like invariants of wreath products, advancing understanding of group generation properties.

Contribution

It classifies all irredundant generating sets of size n-2 in A_n and S_n, and derives formulas for dimension-like invariants of certain wreath products.

Findings

Classified all irredundant generating sets of size n-2 in A_n and S_n

Derived formulas for dimension-like invariants of specific wreath products

Extended previous results on maximum size of irredundant generating sets

Abstract

Whiston proved that the maximum size of an irredundant generating set in the symmetric group $S_n$ is $n-1$, and Cameron and Cara characterized all irredundant generating sets of $S_n$ that achieve this size. Our goal is to extend their results. Using properties of transitive subgroups of the symmetric group, we are able to classify all irredundant generating sets with sizes $n-2$ in both $A_n$ and $S_n$. Next, based on this classification, we derive other interesting properties for the alternating group $A_n$. Finally, using Whiston's lemma, we will derive the formulas for calculating dimension-like invariants of some specific types of wreath products.