In the area of modern algebra known as

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, the Mathieu group ''M₂₂'' is a

sporadic simple group In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...

: 2⁷3²5711 = 443520 : ≈ 4.

History and properties

''M₂₂'' is one of the 26 sporadic groups and was introduced by . It is a 3-fold transitive

permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...

on 22 objects. The

Schur multiplier In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \op ...

of M₂₂ is cyclic of order 12, and the

outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a ...

has order 2. There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature. incorrectly claimed that the Schur multiplier of M₂₂ has order 3, and in a correction incorrectly claimed that it has order 6. This caused an error in the title of the paper announcing the discovery of the

Janko group J4 In the area of modern algebra known as group theory, the Janko group ''J4'' is a sporadic simple group of order : 22133571132329313743 : = 86775571046077562880 : ≈ 9. History ''J4'' is one of the 26 Sporadic groups. Zvoni ...

. showed that the Schur multiplier is in fact cyclic of order 12. calculated the 2-part of all the cohomology of M₂₂.

Representations

M₂₂ has a 3-transitive permutation representation on 22 points, with point stabilizer the group PSL₃(4), sometimes called M₂₁. This action fixes a

Steiner system 250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steine ...

S(3,6,22) with 77 hexads, whose full automorphism group is the automorphism group M₂₂.2 of M₂₂. M₂₂ has three rank 3 permutation representations: one on the 77 hexads with point stabilizer 2⁴:A₆, and two rank 3 actions on 176 heptads that are conjugate under an outer automorphism and have point stabilizer A₇. M₂₂ is the point stabilizer of the action of M₂₃ on 23 points, and also the point stabilizer of the

rank 3 action Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...

of the

Higman–Sims group In the area of modern algebra known as group theory, the Higman–Sims group HS is a sporadic simple group of order : 29⋅32⋅53⋅7⋅11 = 44352000 : ≈ 4. The Schur multiplier has order 2, the outer automorphism ...

on 100 = 1+22+77 points. The triple cover 3.M₂₂ has a 6-dimensional faithful representation over the field with 4 elements. The 6-fold cover of M₂₂ appears in the centralizer 2¹⁺¹².3.(M₂₂:2) of an involution of the

Maximal subgroups

There are no proper subgroups transitive on all 22 points. There are 8 conjugacy classes of maximal subgroups of ''M₂₂'' as follows: * PSL(3,4) or M₂₁, order 20160: one-point stabilizer * 2⁴:A₆, order 5760, orbits of 6 and 16 : Stabilizer of W₂₂ block * A₇, order 2520, orbits of 7 and 15 : There are 2 sets, of 15 each, of simple subgroups of order 168. Those of one type have orbits of 1, 7 and 14; the others have orbits of 7, 8, and 7. * A₇, orbits of 7 and 15 : Conjugate to preceding type in M₂₂:2. * 2⁴:S₅, order 1920, orbits of 2 and 20 (5 blocks of 4) : A 2-point stabilizer in the sextet group * 2³:PSL(3,2), order 1344, orbits of 8 and 14 * M₁₀, order 720, orbits of 10 and 12 (2 blocks of 6) : A one-point stabilizer of M₁₁ (point in orbit of 11) : A non-split

group extension In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\ov ...

of form A₆.2 * PSL(2,11), order 660, orbits of 11 and 11 : Another one-point stabilizer of M₁₁ (point in orbit of 12)

Conjugacy classes

There are 12 conjugacy classes, though the two classes of elements of order 11 are fused under an outer automorphism.

References

* * * * * * Reprinted in * * * * * * * (The title of this paper is incorrect, as the full covering group of M₂₂ was later discovered to be larger: center of order 12, not 6.) * * * * * *

External links