On codes in the projective linear group $PGL(2,q)$

TL;DR

This paper proves a conjecture about the structure of certain codes in the projective linear group PGL(2,q), showing how specific subgroups combine to form the entire group depending on the parity of q.

Contribution

It resolves a conjecture by Green and Liebeck regarding the structure of codes in PGL(2,q), specifically characterizing the group as a product of a dihedral subgroup and a certain set A.

Findings

For even q, the group equals q times the set A times the dihedral subgroup D.

For odd q, the group equals (q-1) times the set A times D.

The structure of codes in PGL(2,q) depends on the parity of q.

Abstract

In this paper, we resolve a conjecture of Green and Liebeck [Disc. Math., 343 (8):117119, 2019] on codes in $PGL(2,q)$. To be specific, we show that: if $D$ is a dihedral subgroup of order $2(q+1)$ in $G=PGL(2,q)$, and $A=\{g\in G: g^{q+1}= 1,\, g^2\ne 1 \}$, then $\lambda G=A\cdot D$, where $\lambda=q$ or $q-1$ according as $q$ is even or odd.