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 McLaughlin group McL is a
sporadic simple group
In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups.
A simpl ...
of
order
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
...
: 898,128,000 = 2
7 ⋅ 3
6 ⋅ 5
3 ⋅ 7 ⋅ 11
: ≈ 9.
History and properties
McL is one of the 26 sporadic groups and was discovered by as an index 2 subgroup of a rank 3 permutation group acting on the
McLaughlin graph with vertices. It fixes a
2-2-3 triangle in the
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Er ...
and thus is a subgroup of the
Conway group
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 auto ...
s
,
, and
. Its
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 \ope ...
has order 3, and its
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 ...
has order 2. The group 3.McL:2 is a maximal subgroup of the
Lyons group
In the area of modern algebra known as group theory, the Lyons group ''Ly'' or Lyons-Sims group ''LyS'' is a sporadic simple group of order
: 51,765,179,004,000,000
: = 283756711313767
: ≈ 5.
History
''Ly'' is one of the 26 ...
.
McL has one conjugacy class of involution (element of order 2), whose centralizer is a maximal subgroup of type 2.A
8. This has a center of order 2; the quotient modulo the center is isomorphic to the alternating group A
8.
Representations
In the
Conway group
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 auto ...
Co
3, McL has the normalizer McL:2, which is maximal in Co
3.
McL has 2 classes of maximal subgroups isomorphic to the
Mathieu group
In group theory, a topic in abstract algebra, 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 objec ...
M
22. An outer automorphism interchanges the two classes of M
22 groups. This outer automorphism is realized on McL embedded as a subgroup of Co
3.
A convenient representation of M
22 is in permutation matrices on the last 22 coordinates; it fixes a 2-2-3 triangle with vertices the origin and the
type 2 points and '. The triangle's edge is
type 3; it is fixed by a Co
3. This M
22 is the
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called a power product or primitive monomial, is a product of powers of variables with n ...
, ''and a maximal'', subgroup of a representation of McL.
(p. 207) shows that the subgroup McL is well-defined. In the
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Er ...
, suppose a type 3 point v is fixed by an instance of
. Count the type 2 points w such that the inner product v·w = 3 (and thus v-w is type 2). He shows their number is and that this Co
3 is transitive on these w.
, McL, = , Co3, /552 = 898,128,000.
McL is the only sporadic group to admit irreducible representations of
quaternionic type. It has 2 such representations, one of dimension 3520 and one of dimension 4752.
Maximal subgroups
found the 12 conjugacy classes of maximal subgroups of McL as follows:
Conjugacy classes
Traces of matrices in a standard 24-dimensional representation of McL are shown. The names of conjugacy classes are taken from the Atlas of Finite Group Representations.
Cycle structures in the rank 3 permutation representation, degree 275, of McL are shown.
Generalized Monstrous Moonshine
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For the
Conway groups, the relevant McKay–Thompson series is
and
.
References
*
Conway, J. H.; Curtis, R. T.;
Norton, S. P.; Parker, R. A.; and
Wilson, R. A.: "''Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups.''" Oxford, England 1985.
*
*
*
*
External links
MathWorld: McLaughlin group Atlas of Finite Group Representations: McLaughlin group
{{DEFAULTSORT:Mclaughlin Group (Mathematics)
Sporadic groups