In
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, a topic in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
, the Mathieu groups are the five
sporadic simple groups
''M''11,
''M''12,
''M''22,
''M''23 and
''M''24 introduced by . They are multiply transitive
permutation groups on 11, 12, 22, 23 or 24 objects. They were the first sporadic groups to be discovered.
Sometimes the notation ''M''
9, ''M''
10, ''M''
20 and ''M''
21 is used for related groups (which act on sets of 9, 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones.
John Conway has shown that one can also extend this sequence up, obtaining the
Mathieu groupoid ''M''13 acting on 13 points. ''M''
21 is simple, but is not a sporadic group, being isomorphic to PSL(3,4).
History
introduced the group ''M''
12 as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) the group ''M''
24, giving its order. In he gave further details, including explicit
generating sets for his groups, but it was not easy to see from his arguments that the groups generated are not just
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
s, and for several years the existence of his groups was controversial. even published a paper mistakenly claiming to prove that ''M''
24 does not exist, though shortly afterwards in he pointed out that his proof was wrong, and gave a proof that the Mathieu groups are simple. finally removed the doubts about the existence of these groups, by constructing them as successive transitive extensions of permutation groups, as well as automorphism groups of
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 Steiner) ...
s.
After the Mathieu groups no new sporadic groups were found until 1965, when the group
J1 was discovered.
Multiply transitive groups
Mathieu was interested in finding multiply transitive permutation groups, which will now be defined. For a natural number ''k'', a permutation group ''G'' acting on ''n'' points is ''k''-transitive if, given two sets of points ''a''
1, ... ''a''
''k'' and ''b''
1, ... ''b''
''k'' with the property that all the ''a''
''i'' are distinct and all the ''b''
''i'' are distinct, there is a group element ''g'' in ''G'' which maps ''a''
''i'' to ''b''
''i'' for each ''i'' between 1 and ''k''. Such a group is called sharply ''k''-transitive if the element ''g'' is unique (i.e. the action on ''k''-tuples is
regular, rather than just transitive).
''M''
24 is 5-transitive, and ''M''
12 is sharply 5-transitive, with the other Mathieu groups (simple or not) being the subgroups corresponding to stabilizers of ''m'' points, and accordingly of lower transitivity (''M''
23 is 4-transitive, etc.). These are the only two 5-transitive groups that are neither
symmetric groups nor
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
s.
[http://www.maths.qmul.ac.uk/~pjc/pps/pps9.pdf ]
The only 4-transitive groups are the
symmetric groups ''S''
''k'' for ''k'' at least 4, the
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
s ''A''
''k'' for ''k'' at least 6, and the Mathieu groups
''M''23 and
''M''11. The full proof requires the
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else ...
, but some special cases have been known for much longer.
It is
a classical result of Jordan that the
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
and
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
s (of degree ''k'' and ''k'' + 2 respectively), and ''M''
12 and ''M''
11 are the only ''sharply'' ''k''-transitive permutation groups for ''k'' at least 4.
Important examples of multiply transitive groups are the
2-transitive group A group G acts 2-transitively on a set S if it acts transitively on the set of distinct ordered pairs \. That is, assuming (without a real loss of generality) that G acts on the left of S, for each pair of pairs (x,y),(w,z)\in S\times S with x \neq ...
s and the
Zassenhaus groups. The Zassenhaus groups notably include the
projective general linear group of a projective line over a finite field, PGL(2,F
''q''), which is sharply 3-transitive (see
cross ratio) on
elements.
Order and transitivity table
Constructions of the Mathieu groups
The Mathieu groups can be constructed in various ways.
Permutation groups
''M''
12 has a simple subgroup of order 660, a maximal subgroup. That subgroup is isomorphic to the
projective special linear group PSL
2(F
11) over the
field of 11 elements. With −1 written as a and infinity as b, two standard generators are (0123456789a) and (0b)(1a)(25)(37)(48)(69). A third generator giving ''M''
12 sends an element ''x'' of F
11 to 4''x''
2 − 3''x''
7; as a permutation that is (26a7)(3945).
This group turns out not to be isomorphic to any member of the infinite families of finite simple groups and is called sporadic. ''M''
11 is the stabilizer of a point in ''M''
12, and turns out also to be a sporadic simple group. ''M''
10, the stabilizer of two points, is not sporadic, but is an
almost simple group In mathematics, a group is said to be almost simple if it contains a non- abelian simple group and is contained within the automorphism group of that simple group – that is, if it fits between a (non-abelian) simple group and its automorphism gro ...
whose
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
is the
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic pr ...
A
6. It is thus related to the
exceptional outer automorphism of A
6. The stabilizer of 3 points is the
projective special unitary group PSU(3,2
2), which is solvable. The stabilizer of 4 points is the
quaternion group.
Likewise, ''M''
24 has a maximal simple subgroup of order 6072 isomorphic to PSL
2(F
23). One generator adds 1 to each element of the field (leaving the point ''N'' at infinity fixed), i. e. (0123456789ABCDEFGHIJKLM)(''N''), and the other is the
order reversing permutation, (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI). A third generator giving ''M''
24 sends an element ''x'' of F
23 to 4''x''
4 − 3''x''
15 (which sends perfect squares via
and non-perfect squares via
); computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF).
The stabilizers of 1 and 2 points, ''M''
23 and ''M''
22 also turn out to be sporadic simple groups. The stabilizer of 3 points is simple and isomorphic to the
projective special linear group PSL
3(4).
These constructions were cited by . ascribe the permutations to Mathieu.
Automorphism groups of Steiner systems
There exists
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
equivalence a unique ''S''(5,8,24)
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 Steiner) ...
W
24 (the
Witt design). The group ''M''
24 is the automorphism group of this Steiner system; that is, the set of permutations which map every block to some other block. The subgroups ''M''
23 and ''M''
22 are defined to be the stabilizers of a single point and two points respectively.
Similarly, there exists up to equivalence a unique ''S''(5,6,12) Steiner system W
12, and the group ''M''
12 is its automorphism group. The subgroup ''M''
11 is the stabilizer of a point.
''W''
12 can be constructed from the
affine geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of ''parallel lines'' is one of the main properties that is ind ...
on the
vector space , an ''S''(2,3,9) system.
An alternative construction of ''W''
12 is the 'Kitten' of .
An introduction to a construction of ''W''
24 via the
Miracle Octad Generator of R. T. Curtis and Conway's analog for ''W''
12, the miniMOG, can be found in the book by Conway and
Sloane.
Automorphism groups on the Golay code
The group ''M''
24 is the
permutation automorphism group of the
extended binary Golay code ''W'', i.e., the group of permutations on the 24 coordinates that map ''W'' to itself. All the Mathieu groups can be constructed as groups of permutations on the binary Golay code.
''M''
12 has index 2 in its automorphism group, and ''M''
12:2 happens to be isomorphic to a subgroup of ''M''
24. ''M''
12 is the stabilizer of a dodecad, a codeword of 12 1's; ''M''
12:2 stabilizes a partition into 2 complementary dodecads.
There is a natural connection between the Mathieu groups and the larger
Conway groups
In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by .
The largest of the Conway groups, Co0, is the group of autom ...
, because the
Leech lattice was constructed on the binary Golay code and in fact both lie in spaces of dimension 24. The Conway groups in turn are found in the
Monster group.
Robert Griess
Robert Louis Griess, Jr. (born 1945, Savannah, Georgia) is a mathematician working on finite simple groups and vertex algebras. He is currently the John Griggs Thompson Distinguished University Professor of mathematics at University of Michi ...
refers to the 20 sporadic groups found in the Monster as the Happy Family, and to the Mathieu groups as the first generation.
Dessins d'enfants
The Mathieu groups can be constructed via
dessins d'enfants
In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French for ...
, with the dessin associated to ''M''
12 suggestively called "Monsieur Mathieu" by .
References
*
*
*
*
* Reprinted in
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* (an introduction for the lay reader, describing the Mathieu groups in a historical context)
*
*
*
External links
ATLAS: Mathieu group ''M''10ATLAS: Mathieu group ''M''11ATLAS: Mathieu group ''M''12ATLAS: Mathieu group ''M''20ATLAS: Mathieu group ''M''21ATLAS: Mathieu group ''M''22ATLAS: Mathieu group ''M''23ATLAS: Mathieu group ''M''24*
*
Scientific AmericanA set of puzzles based on the mathematics of the Mathieu groups
Sporadic M12 An iPhone app that implements puzzles based on ''M''
12, presented as one "spin" permutation and a selectable "swap" permutation
{{DEFAULTSORT:Mathieu Group
Sporadic groups