Mathieu group M23
{{Short description|Sporadic simple group}}
{{DISPLAYTITLE:Mathieu group M23}}
{{for|general background and history of the Mathieu sporadic groups|Mathieu group}}
{{Group theory sidebar |Finite}}
In the area of modern algebra known as group theory, the Mathieu group M23 is a sporadic simple group of order
: 10,200,960 = 27{{·}}32{{·}}5{{·}}7{{·}}11{{·}}23
: ≈ 1 × 107.
History and properties
M23 is one of the 26 sporadic groups and was introduced by {{harvs|txt |authorlink=Émile Léonard Mathieu |last=Mathieu |year1=1861 |year2=1873}}. It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial.
{{harvtxt|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 {{math|F211}} be the finite field with 211 elements. Its group of units has order {{math|211}} − 1 = 2047 = 23 · 89, so it has a cyclic subgroup {{math|C}} of order 23.
The Mathieu group M23 can be identified with the group of {{math|F2}}-linear automorphisms of {{math|F211}} that stabilize {{math|C}}. More precisely, the action of this automorphism group on {{math|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:
| class="wikitable"
|+ Maximal subgroups of M23 | ||||
| No. | Structure | Order | Index | Comments |
|---|---|---|---|---|
| 1 | M22 | align=right|443,520 = 27·32·5·7·11 | align=right| 23 | point stabilizer |
| 2 | L3(4):2 | align=right| 40,320 = 27·32·5·7 | align=right| 253 = 11·23 | has orbits of sizes 21 and 2 |
| 3 | 24:A7 | align=right| 40,320 = 27·32·5·7 | align=right| 253 = 11·23 | has orbits of sizes 7 and 16; stabilizer of W23 block |
| 4 | A8 | align=right| 20,160 = 26·32·5·7 | align=right| 506 = 2·11·23 | has orbits of sizes 8 and 15 |
| 5 | M11 | align=right| 7,920 = 24·32·5·11 | align=right| 1,288 = 23·7·23 | has orbits of sizes 11 and 12 |
| 6 | (24:A5):S3 ≅ M20:S3 | align=right| 5,760 = 27·32·5 | align=right| 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 | align=right| 253 = 11·23 | align=right|40,320 = 27·32·5·7 | simply transitive |
Conjugacy classes
| class="wikitable" style="margin: 1em auto;" | |||
| 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 | |
| rowspan="2" | 7 = 7 | 728640 = 26 · 32 · 5 · 11 · 23 | 1273 | rowspan="2"|power equivalent |
| 728640 = 26 · 32 · 5 · 11 · 23 | 1273 | ||
| 8 = 23 | 1275120 = 24 · 32 · 5 · 7 · 11 · 23 | 1·2·4·82 | |
| rowspan="2" | 11 = 11 | 927360= 27 · 32 · 5 · 7 · 23 | 1·112 | rowspan="2"|power equivalent |
| 927360= 27 · 32 · 5 · 7 · 23 | 1·112 | ||
| rowspan="2" | 14 = 2 · 7 | 728640= 26 · 32 · 5 · 11 · 23 | 2·7·14 | rowspan="2"|power equivalent |
| 728640= 26 · 32 · 5 · 11 · 23 | 2·7·14 | ||
| rowspan="2" | 15 = 3 · 5 | 680064= 27 · 3 · 7 · 11 · 23 | 3·5·15 | rowspan="2"|power equivalent |
| 680064= 27 · 3 · 7 · 11 · 23 | 3·5·15 | ||
| rowspan="2" | 23 = 23 | 443520= 27 · 32 · 5 · 7 · 11 | 23 | rowspan="2"|power equivalent |
| 443520= 27 · 32 · 5 · 7 · 11 | 23 |
References
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External links
- [http://mathworld.wolfram.com/MathieuGroups.html MathWorld: Mathieu Groups]
- [http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M23/ Atlas of Finite Group Representations: M23]
{{DEFAULTSORT:Mathieu Group M23}}