The Mathieu group $M_{23}$ as additive functions on the finite field of size $2^{11}$

Abstract

We explicitly extend the standard permutation action of the Mathieu group $M_{23}$ on a 23 element set $C=C_{23}$ contained in a finite field of $2^{11}$ elements ${\mathbb{F}}_{2^{11}}$ to additive functions on this finite field. That is we represent $M_{23}$ as functions $\ddot{I}†:{\mathbb{F}}_{2^{11}}\to {\mathbb{F}}_{2^{11}}$ such that $\ddot{I}†(x+y)=\ddot{I}†(x)+\ddot{I}†(y)$ and $\ddot{I}†{|}_C$ is the standard permutation action. We give explicit $11\times 11$ matrices for the pair of standard generators of order $23$ and order $5$, as well as many tables to help facilitate future calculations.

15 pages