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Groups having minimal covering number 2 of diagonal type

Marco Fusari, Andrea Previtali, Pablo Spiga

Abstract

Garonzi and Lucchini~\cite{GL} explored finite groups $G$ possessing a normal $2$-covering, where no proper quotient of $G$ exhibits such a covering. Their investigation offered a comprehensive overview of these groups, delineating that such groups fall into distinct categories: almost simple, affine, product action, or diagonal. In this paper, we focus on the family falling under the diagonal type. Specifically, we present a thorough classification of finite diagonal groups possessing a normal $2$-covering, with the attribute that no proper quotient of $G$ has such a covering.

Groups having minimal covering number 2 of diagonal type

Abstract

Garonzi and Lucchini~\cite{GL} explored finite groups possessing a normal -covering, where no proper quotient of exhibits such a covering. Their investigation offered a comprehensive overview of these groups, delineating that such groups fall into distinct categories: almost simple, affine, product action, or diagonal. In this paper, we focus on the family falling under the diagonal type. Specifically, we present a thorough classification of finite diagonal groups possessing a normal -covering, with the attribute that no proper quotient of has such a covering.
Paper Structure (3 sections, 6 theorems, 44 equations)

This paper contains 3 sections, 6 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Groups of diagonal type
  3. Proof of Theorem $\ref{['thrm:1']}$

Key Result

Theorem 1.2

Let $G$ be a group with $\gamma(G)=2$ and $\gamma(G/N)>2$, for every non-identity normal subgroup $N$ of $G$. Let $H$ and $K$ be maximal subgroups of $G$ witnessing that $\gamma(G)=2$; moreover, in view of GL, let $H$ be the component containing the socle of $G$. If $G$ is of diagonal type, then $G,

Theorems & Definitions (12)

  • Example 1.1
  • Theorem 1.2
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Theorem $\ref{['thrm:1']}$
  • Lemma 3.4
  • ...and 2 more