Covering certain monolithic groups with proper subgroups

TL;DR

This paper investigates the minimal number of proper subgroups needed to cover certain finite groups with a unique minimal normal subgroup, providing bounds and exact values for specific cases.

Contribution

It establishes bounds for the covering number (G) in groups with a unique minimal normal subgroup isomorphic to a direct power of A_n, and computes an exact value for a specific wreath product.

Findings

Bounds for (G) in groups with normal subgroup A_n^m

Exact value (A_5 \u2297 C_2) = 57

Analysis of subgroup coverings in specific group structures

Abstract

Given a finite non-cyclic group $G$, call $\sigma(G)$ the least number of proper subgroups of $G$ needed to cover $G$. In this paper we give lower and upper bounds for $\sigma(G)$ for $G$ a group with a unique minimal normal subgroup $N$ isomorphic to $A_n^m$ where $n \geq 5$ and $G/N$ is cyclic. We also show that $\sigma(A_5 \wr C_2)=57$.