# Linear bounds for the normal covering number of the symmetric and alternating groups

## TL;DR

This paper establishes linear lower bounds for the normal covering number of symmetric and alternating groups, providing insights into their subgroup structures and element coverage.

## Contribution

It introduces new linear bounds for the normal covering number of symmetric and alternating groups, advancing understanding of their subgroup coverings.

## Key findings

- Linear bounds for $oldsymbol{\gamma(S_n)}$ when n is even.
- Linear bounds for $oldsymbol{\gamma(A_n)}$ when n is odd.
- Enhanced understanding of subgroup coverings in symmetric and alternating groups.

## Abstract

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the minimum number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We find lower bounds linear in $n$ for $\gamma(S_n)$, when $n$ is even, and for $\gamma(A_n)$, when $n$ is odd.