# The covering radius of $\mathrm{PGL}_2(q)$

## TL;DR

This paper determines the covering radius of the finite projective general linear groups, specifically $ ext{PGL}_2(q)$, showing it depends on the parity of q, with exact values provided.

## Contribution

It provides the first explicit calculation of the covering radius of $ ext{PGL}_2(q)$, a significant result in the study of permutation groups.

## Key findings

- Covering radius of $ ext{PGL}_2(q)$ is $q-2$ for even q.
- Covering radius of $ ext{PGL}_2(q)$ is $q-3$ for odd q.
- Exact values depend on the parity of q.

## Abstract

The covering radius of a subset $C$ of the symmetric group $\mathrm{S}_n$ is the maximal Hamming distance of an element of $\mathrm{S}_n$ from $C$. This note determines the covering radii of the finite projective general linear groups. It turns out that the covering radius of $\mathrm{PGL}_2(q)$ is $q-2$ if $q$ is even, and is $q-3$ if $q$ is odd.