Mathieu group M₂₃

In the area of modern algebra known as group theory, the Mathieu group M₂₃ is a sporadic simple group of order

2⁷ · 3² · 5 · 7 · 11 · 23 = 10200960

≈ 1 × 10⁷.

History and properties

M₂₃ is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial.

Milgram (2000) calculated the integral cohomology, and showed in particular that M₂₃ has the unusual property that the first 4 integral homology groups all vanish.

The inverse Galois problem seems to be unsolved for M₂₃. In other words, no polynomial in Z[x] seems to be known to have M₂₃ as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups.

Construction using finite fields

Let $F 211$ be the finite field with 2¹¹ elements. Its group of units has order $211$ − 1 = 2047 = 23 · 89, so it has a cyclic subgroup $C$ of order 23.

The Mathieu group M₂₃ can be identified with the group of $F 2$ -linear automorphisms of $F 211$ that stabilize $C$ . More precisely, the action of this automorphism group on $C$ can be identified with the 4-fold transitive action of M₂₃ on 23 objects.

Representations

M₂₃ is the point stabilizer of the action of the Mathieu group M24 on 24 points, giving it a 4-transitive permutation representation on 23 points with point stabilizer the Mathieu group M22.

M₂₃ has 2 different rank 3 actions on 253 points. One is the action on unordered pairs with orbit sizes 1+42+210 and point stabilizer M₂₁.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 2⁴.A₇.

The integral representation corresponding to the permutation action on 23 points decomposes into the trivial representation and a 22-dimensional representation. The 22-dimensional representation is irreducible over any field of characteristic not 2 or 23.

Over the field of order 2, it has two 11-dimensional representations, the restrictions of the corresponding representations of the Mathieu group M24.

Maximal subgroups

There are 7 conjugacy classes of maximal subgroups of M₂₃ as follows:

M₂₂, order 443520
PSL(3,4):2, order 40320, orbits of 21 and 2
2⁴:A₇, order 40320, orbits of 7 and 16

Stabilizer of W₂₃ block

A₈, order 20160, orbits of 8 and 15
M₁₁, order 7920, orbits of 11 and 12
(2⁴:A₅):S₃ or M₂₀:S₃, order 5760, orbits of 3 and 20 (5 blocks of 4)

One-point stabilizer of the sextet group

23:11, order 253, simply transitive

Conjugacy classes

Order	No. elements	Cycle structure
1 = 1	1	1²³
2 = 2	3795 = 3 · 5 · 11 · 23	1⁷2⁸
3 = 3	56672 = 2⁵ · 7 · 11 · 23	1⁵3⁶
4 = 2²	318780 = 2² · 3² · 5 · 7 · 11 · 23	1³2²4⁴
5 = 5	680064 = 2⁷ · 3 · 7 · 11 · 23	1³5⁴
6 = 2 · 3	850080 = 2⁵ · 3 · 5 · 7 · 11 · 23	1·2²3²6²
7 = 7	728640 = 2⁶ · 3² · 5 · 11 · 23	1²7³	power equivalent
7 = 7	728640 = 2⁶ · 3² · 5 · 11 · 23	1²7³	power equivalent
8 = 2³	1275120 = 2⁴ · 3² · 5 · 7 · 11 · 23	1·2·4·8²
11 = 11	927360= 2⁷ · 3² · 5 · 7 · 23	1·11²	power equivalent
11 = 11	927360= 2⁷ · 3² · 5 · 7 · 23	1·11²	power equivalent
14 = 2 · 7	728640= 2⁶ · 3² · 5 · 11 · 23	2·7·14	power equivalent
14 = 2 · 7	728640= 2⁶ · 3² · 5 · 11 · 23	2·7·14	power equivalent
15 = 3 · 5	680064= 2⁷ · 3 · 7 · 11 · 23	3·5·15	power equivalent
15 = 3 · 5	680064= 2⁷ · 3 · 7 · 11 · 23	3·5·15	power equivalent
23 = 23	443520= 2⁷ · 3² · 5 · 7 · 11	23	power equivalent
23 = 23	443520= 2⁷ · 3² · 5 · 7 · 11	23	power equivalent

References

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