Sporadic simple group
In the area of modern algebra known as group theory , the Mathieu group M 23 is a sporadic simple group of order
27 · 32 · 5 · 7 · 11 · 23 = 10200960
≈ 1 × 107 . History and properties [ edit ] M 23 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 [ edit ] 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.
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 .
There are 7 conjugacy classes of maximal subgroups of M 23 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
Order
No. elements
Cycle structure
1 = 1
1
123
2 = 2
3795 = 3 · 5 · 11 · 23
17 28
3 = 3
56672 = 25 · 7 · 11 · 23
15 36
4 = 22
318780 = 22 · 32 · 5 · 7 · 11 · 23
13 22 44
5 = 5
680064 = 27 · 3 · 7 · 11 · 23
13 54
6 = 2 · 3
850080 = 25 · 3 · 5 · 7 · 11 · 23
1·22 32 62
7 = 7
728640 = 26 · 32 · 5 · 11 · 23
12 73
power equivalent
728640 = 26 · 32 · 5 · 11 · 23
12 73
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
927360= 27 · 32 · 5 · 7 · 23
1·112
14 = 2 · 7
728640= 26 · 32 · 5 · 11 · 23
2·7·14
power equivalent
728640= 26 · 32 · 5 · 11 · 23
2·7·14
15 = 3 · 5
680064= 27 · 3 · 7 · 11 · 23
3·5·15
power equivalent
680064= 27 · 3 · 7 · 11 · 23
3·5·15
23 = 23
443520= 27 · 32 · 5 · 7 · 11
23
power equivalent
443520= 27 · 32 · 5 · 7 · 11
23
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