Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generation of Iterated Wreath Products Constructed from Almost Simple Groups

Jiaping Lu

TL;DR

This paper addresses the minimal generator problem for iterated wreath products constructed from almost simple finite groups via the regular action. It extends Lucchini's framework for wreath products with simple groups to the almost simple setting by introducing invariants such as $\delta_G(M)$, $h_G(M)$, and the presentation rank, and uses them to analyze $d(A\operatorname{wr}G)$ where $A$ is abelian. The main result gives a closed-form expression for $d(W)$, with $W=G_k\wr\cdots\wr G_1$, as $d(W)=\max_p(d(A\times G_1), d(A)+1, d_p(A)+2)$, where $A=(G_k/G_k')\times\cdots\times(G_2/G_2')$ and the maximum runs over primes $p$ dividing both $|A|$ and $|S|$ (the socle of $G_1$). The paper handles Lie-type, alternating, and sporadic cases, showing that in the latter two the result reduces to Lucchini’s original formulation, while for Lie-type factors the analysis hinges on the proposed invariants to yield the exact generator bound. Overall, the work advances understanding of generation properties in complex wreath-product constructions and provides precise, implementable bounds for finite group theory and related computations.

Abstract

Let G1, G2, ... be a sequence of almost simple groups and construct a sequence (Wi) of wreath products via W1 = G1 and, for each i > 1, Wi+1 = Gi+1 wr Wi via the regular action of each Gi. We determine the minimum number d(Wi) of generators required for each wreath product in this sequence.

Generation of Iterated Wreath Products Constructed from Almost Simple Groups

TL;DR

This paper addresses the minimal generator problem for iterated wreath products constructed from almost simple finite groups via the regular action. It extends Lucchini's framework for wreath products with simple groups to the almost simple setting by introducing invariants such as , , and the presentation rank, and uses them to analyze where is abelian. The main result gives a closed-form expression for , with , as , where and the maximum runs over primes dividing both and (the socle of ). The paper handles Lie-type, alternating, and sporadic cases, showing that in the latter two the result reduces to Lucchini’s original formulation, while for Lie-type factors the analysis hinges on the proposed invariants to yield the exact generator bound. Overall, the work advances understanding of generation properties in complex wreath-product constructions and provides precise, implementable bounds for finite group theory and related computations.

Abstract

Let G1, G2, ... be a sequence of almost simple groups and construct a sequence (Wi) of wreath products via W1 = G1 and, for each i > 1, Wi+1 = Gi+1 wr Wi via the regular action of each Gi. We determine the minimum number d(Wi) of generators required for each wreath product in this sequence.
Paper Structure (6 sections, 10 theorems, 36 equations)

This paper contains 6 sections, 10 theorems, 36 equations.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Acknowledgements:
  4. Preliminaries
  5. Wreath products of an abelian group by an almost simple group of Lie type
  6. Proof of main theorem

Key Result

Theorem 1.1

Let $G_{1}$, $G_{2}$, …, $G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let be the iterated wreath product constructed via the regular action of each factor. Set Then where $p$ ranges over the set of prime numbers dividing both $\mathopen{|}A\mathclose{|}$ and $\mathopen{|}S\mathclose{|}$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 2.1: Lucchini, Menegazzo LuMe
  • Lemma 2.2: Roggenkamp Roggenkamp
  • Lemma 2.3: Gruenberg Gru76
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 4.1
  • Lemma 4.2