In the area of

modern algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...

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 ''M''₂₃ is a sporadic simple group of order : 2⁷3²571123 = 10200960 : ≈ 1 × 10⁷.

History and properties

''M''₂₃ is one of the 26 sporadic groups and was introduced by . It is a 4-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 it ...

on 23 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 \oper ...

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 t ...

are both

trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...

. 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 In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There ...

seems to be unsolved for M₂₃. In other words, no

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

in Z 'x''seems to be known to have M₂₃ as its

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

. The inverse Galois problem is solved for all other sporadic simple groups.

Construction using finite fields

Let be the

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

with 2¹¹ elements. Its group of units has order − 1 = 2047 = 23 · 89, so it has a

cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

of order 23. The Mathieu group M₂₃ can be identified with the group of - linear automorphisms of that stabilize . More precisely, the

action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...

of this

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

on 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 In the area of modern algebra known as group theory, the Mathieu group ''M24'' is a sporadic simple group of order : 21033571123 = 244823040 : ≈ 2. History and properties ''M24'' is one of the 26 sporadic groups and was ...

on 24 points, giving it a 4-transitive

permutation representation In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices. The term also refers ...

on 23 points with point stabilizer the

Mathieu group M22 In the area of modern algebra known as group theory, the Mathieu group ''M22'' is a sporadic simple group of Order (group theory), order : 27325711 = 443520 : ≈ 4. History and properties ''M22'' is one of the 26 sporadic gro ...

. M₂₃ has 2 different

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 * ...

s 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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

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

Maximal subgroups

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

Conjugacy classes

References

* * * Reprinted in * * * * * * * * * * *

External links

MathWorld: Mathieu Groups

Atlas of Finite Group Representations: M₂₃
{{DEFAULTSORT:Mathieu Group M23 Sporadic groups