# Mathieu group $M_{23}$ as an algebraic group via additive polynomials

An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is the group of additive maps of $F$ to itself which permute the set $C$. The restriction to $C$ then is the natural action of $M_{23}$ of degree $23$.

Any additive map of $F$ to itself has the form $x\mapsto\sum_{i=0}^{10}a_ix^{2^i}$ for $a_i\in F$. The condition that such maps preserve $C$ can be expressed easily as an equivalent set of polynomial conditions on the coefficients $a_0,\dots,a_{10}$. For instance, for a variable $t$, compare coefficients of $$\prod_{c\in C}(t-\sum_{i=0}^{10}a_ic^{2^i})=t^{23}-1.$$ In addition, we need the equations $a_i^{2^{11}}=a_i$ to ensure that the $a_i$ are in $F$.

However, the polynomials we obtain this way (or some other way, there are somewhat smarter possibilities) are extremely complicated.

Of course, any finite subgroup of a linear group is an algebraic set, that is a solution of a system of polynomial equations. And in general such a system will be messy.

Here, I somehow expect that there should be a simple system of polynomials discribing $M_{23}$ as an algebraic set. The obvious attempt would be to first take a system as above, and then hope that a Groebner basis looks better. After some naive attempts it seems that the systems are too complex for Magma or Singular to compute the Groebner bases.

Before trying to refine the approach, here is my question: Has anyone seen this version to describe $M_{23}$?

• This looks like a reasonable (though maybe quite difficult) question. But I wonder what consequences such a description would have? Mar 13, 2017 at 14:18
• the natural object preserved by the group here is dual of the perfect binary Golay code en.wikipedia.org/wiki/Binary_Golay_code One might ask for similar descriptions of smaller codes to begin with. Mar 13, 2017 at 15:16
• E.g. one may ask for such a description for PSL(3,2): I suppose you have to replace 11 by 3 and 23 by 7, i.e. $|C|=7$ and $F=\mathbb{F}_{2^3}$, and consider PSL(3,2) as the group of additive maps of $F$ preserving $C$. (Disclaimer---I didn't check all the details here :-)). Mar 13, 2017 at 15:29
• This is a very old question, but I can't help adding the following comment, that may or may not be relevant. I wonder whether Abhyankar's paper, "Mathieu group coverings in characteristic two" is relevant here? He demonstrates that there are very innocuous looking degree 23 trinomials over ${\mathbb F}_2(X)$ with Galois group $M_{23}$. His example is $Y^{23}+XY^3+1$. Mar 2, 2023 at 20:48
• @DaveBenson Indeed, I had also tried something along this idea. The upper bound of the Galois group of $Y^{23}+XY^3+1$ is achieved by showing that this polynomial divides an additive polynomial of degree $2^{11}$. So actually this additive polynomial has Galois group $M_{23}$, and its groups acts on an $11$-dimensional $\mathbb F_2$-space. Apr 4, 2023 at 21:09