The description of the Mathieu group ( M_{23} ) as the group of additive (linearized) polynomials over ( \mathbb{F}_{2^{11}} ) that permute a multiplicative subgroup ( C ) of order 23 does not appear to be a standard or widely referenced characterization in the existing literature on finite simple groups or algebraic geometry.

Mathieu groups, including ( M_{23} ), are typically construct...

Yes — this is a classical and widely used model of M23.

What you are doing can be phrased geometrically as follows. Identify the 11‑dimensional F2–vector space V with the field F = GF(2^11) under addition, and let C ≤ F× be the unique multiplicative subgroup of order 23. Regard C as a 23‑subset of the projective space PG(10,2) (via V{0} modulo scalars). Then the setwise stabilizer in GL(11,2) of...

Yes – the realisation of the Mathieu groups inside the group AΓL(1,2m) is classical and in the case m = 11 it gives exactly the group M23 that you wrote down. A convenient reference is

Witt, E.:  Die 5-fach transitiven Gruppen von Mathieu,  
Abh. Math. Seminar Hamburg 12 (1938), 256 – 264.  

Witt shows that

 G = { F2–linear ( = “additive’’ ) maps L : F2 ^11 → F2 ^11 for which L(C)=C }...

This is a fascinating question about finding a nice algebraic description of the Mathieu group using additive polynomials over finite fields.

The construction you describe is indeed elegant: consists of additive maps (where ) that permute the cyclic group of order 23 in . Since additive maps have the form $x \mapsto \sum_{i=0}^{10} a_i...

An excellent and very insightful question. The description of as the group of additive permutations of the 23rd roots of unity in is indeed a classic and elegant construction. You are right to suspect that there might be a simpler way to express the conditions on the coefficients than the direct comparison of elementary symmetric polynomials.

The Simplifie...

<think> We are given a description of the Mathieu group \( M_{23} \) as the group of additive maps of the finite field \( \mathbb{F}_{2^{11}} \) to itself that permute the multiplicative subgroup \( C \) of order 23. The additive maps are of the form \( x \mapsto \sum_{i=0}^{10} a_i x^{2^i} \) for \( a_i \in \mathbb{F}_{2^{11}} \). The condition that such a map preserves \( C \) is equivalent to t...

Nothing “nice’’ has ever turned up in the literature beyond the naïve linear‐polynomial cut‐out that you describe. In fact if one identifies F = 𝔽_{2^{11}} ≅ 𝔽₂[x]/(m(x)) and writes an additive endomorphism of F as

f(x) = a₀x + a₁x² + a₂x^{2²} + … + a_{10}x^{2^{10}},

then the condition “f permutes the cyclic subgroup C of order 23” is equivalent to

(1) a_i^{2^{11}} − a_i = 0, ...

Mathieu Group as an Algebraic Group via Additive Polynomials

The description of the Mathieu group that you present—as the group of additive maps on preserving a multiplicative subgroup of order 23—is indeed an elegant formulation, but I understand your frustration with the complexity of the resulting polynomial conditions.

This representation of ...

Yes – the description you are looking for is classical. It goes back to Witt’s first construction of the Mathieu groups in 1938 and has been re-worked many times since (see the references at the end). The up-shot is that the set of “linearised’’ (i.e. additive) polynomials

    L(x)=a0x + a1x^2 + … + a10x^{2^10}     (ai∈F:=F2^11)

which permute the 23-element subgroup

    C = { 0 ≠ c ∈ ...