2-covering numbers of some finite solvable groups

TL;DR

This paper investigates $2$-coverings of finite groups and the associated invariant $\sigma_2(G)$. It disproves a conjecture asserting that for finite solvable groups with $d(G)\ge3$, $\sigma_2(G)$ must have the form $p^{2t}+p^{t}+1$. By constructing solvable groups with $\sigma_2(G)=1+p^2+p^3+p^4$ for every odd prime $p$, and $\sigma_2(G)=q^2+q^3+q^4+p$ when a prime $p$ divides $q+1$, the work shows $\sigma_2(G)$ can take infinitely many even values. The approach builds semidirect products $G=V^{r+1}\rtimes H$ with $d(H)=2$ acting on a faithful irreducible $H$-module $V$, analyzes two types of maximal subgroups, and uses generation properties of $H$-modules to determine $\sigma_2(G)$ via a two-case analysis yielding $\gamma+1$ or $\gamma+p$, where $\gamma$ counts the first-type maxima. These results provide explicit counterexamples and clarify how the $2$-covering number can vary in solvable groups, with potential implications for related covering problems in finite group theory.

Abstract

A 2-covering for a finite group $G$ is a set of proper subgroups of $G$ such that every pair of elements of $G$ is contained in at least one subgroup in the set. The minimal number of subgroups needed to 2-cover a group $G$ is called the 2-covering number and denoted by $σ_2(G).$ In \cite{gk} it is conjectured that if $G$ is solvable and not 2-generated, then $σ_2(G)=1+q+q^2,$ where $q$ is a prime power. We disprove this conjecture.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• proof : Proof of Theorem \ref{['uno']}
• proof : Proof of Theorem \ref{['due']}