Generation of iterated wreath products constructed from alternating, symmetric and cyclic groups

TL;DR

This paper investigates the minimal number of generators needed for a sequence of wreath products formed from alternating, symmetric, and cyclic groups, providing insights into their algebraic structure.

Contribution

It determines the minimum number of generators for each wreath product in a sequence constructed from specific types of groups.

Findings

Explicit formulas for the minimal number of generators for each wreath product.

Characterization of how the number of generators evolves in the sequence.

Insights into the algebraic complexity of iterated wreath products.

Abstract

Let $G_{1}$, $G_{2}$, ... be a sequence of groups each of which is either an alternating group, a symmetric group or a cyclic group and construct a sequence $(W_{i})$ of wreath products via $W_{1} = G_{1}$ and, for each $i \geq 1$, $W_{i+1} = G_{i+1} \operatorname{wr} G_{i}$ via the natural permutation action. We determine the minimum number $d(W_{i})$ of generators required for each wreath product in this sequence.