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Mathieu group M23

Sporadic simple group

In the area of modern algebra known as group theory, the Mathieu group M23 is a sporadic simple group of order : 10,200,960 = 2732571123 : ≈ 1 × 107.

History and properties

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

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 F211 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 F2-linear automorphisms of F211 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:

No.	Structure	Order
1	M22	443,520
= 27·32·5·7·11	23	point stabilizer
2	L3(4):2	40,320
= 27·32·5·7	253
= 11·23	has orbits of sizes 21 and 2
3	24:A7	40,320
= 27·32·5·7	253
= 11·23	has orbits of sizes 7 and 16; stabilizer of W23 block
4	A8	20,160
= 26·32·5·7	506
= 2·11·23	has orbits of sizes 8 and 15
5	M11	7,920
= 24·32·5·11	1,288
= 23·7·23	has orbits of sizes 11 and 12
6	(24:A5):S3 ≅ M20:S3	5,760
= 27·32·5	1,771
= 7·11·23	has orbits of sizes 3 and 20 (5 blocks of 4); one-point stabilizer of the sextet group
7	23:11	253
= 11·23	40,320
= 27·32·5·7	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
728640 = 26 · 32 · 5 · 11 · 23	1273
8 = 23	1275120 = 24 · 32 · 5 · 7 · 11 · 23	1·2·4·82
11 = 11	927360= 27 · 32 · 5 · 7 · 23	1·112
927360= 27 · 32 · 5 · 7 · 23	1·112
14 = 2 · 7	728640= 26 · 32 · 5 · 11 · 23	2·7·14
728640= 26 · 32 · 5 · 11 · 23	2·7·14
15 = 3 · 5	680064= 27 · 3 · 7 · 11 · 23	3·5·15
680064= 27 · 3 · 7 · 11 · 23	3·5·15
23 = 23	443520= 27 · 32 · 5 · 7 · 11	23
443520= 27 · 32 · 5 · 7 · 11	23

References

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