McLaughlin group (mathematics)
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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 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 ...
of order :   27 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 = 898,128,000 : ≈ 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 E ...
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 \mathrm_0, \mathrm_2, and \mathrm_3. 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 \oper ...
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 a t ...
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 :    283756711313767 : = 51765179004000000 : ≈ 5. History ''Ly'' is one of the 26 spor ...
. McL has one conjugacy class of involution (element of order 2), whose centralizer is a maximal subgroup of type 2.A8. This has a center of order 2; the quotient modulo the center is isomorphic to the alternating group A8.


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 ...
Co3, McL has the normalizer McL:2, which is maximal in Co3. 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 ...
M22. An outer automorphism interchanges the two classes of M22 groups. This outer automorphism is realized on McL embedded as a subgroup of Co3. A convenient representation of M22 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 Co3. This M22 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 power product, is a product of powers of variables with nonnegative integer exponent ...
, ''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 E ...
, suppose a type 3 point v is fixed by an instance of \mathrm_3. 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 Co3 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: * U4(3) order 3,265,920 index 275 – point stabilizer of its action on the McLaughlin graph * M22 order 443,520 index 2,025 (two classes, fused under an outer automorphism) * U3(5) order 126,000 index 7,128 * 31+4:2.S5 order 58,320 index 15,400 * 34: M10 order 58,320 index 15,400 * L3(4):22 order 40,320 index 22,275 * 2.A8 order 40,320 index 22,275 – centralizer of involution * 24:A7 order 40,320 index 22,275 (two classes, fused under an outer automorphism) * M11 order 7,920 index 113,400 * 5+1+2:3:8 order 3,000 index 299,376


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 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 ...
, the relevant McKay–Thompson series is T_(\tau) and T_(\tau).


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