McLaughlin sporadic group

In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order

2⁷ ⋅ 3⁶ ⋅ 5³ ⋅ 7 ⋅ 11 = 898,128,000

≈ 9×10⁸.

History and properties

McL is one of the 26 sporadic groups and was discovered by Jack McLaughlin (1969) as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with 275 = 1 + 112 + 162 vertices. It fixes a 2-2-3 triangle in the Leech lattice and thus is a subgroup of the Conway groups $\mathrm {Co} _{0}$ , $\mathrm {Co} _{2}$ , and $\mathrm {Co} _{3}$ . Its Schur multiplier has order 3, and its outer automorphism group has order 2. The group 3.McL:2 is a maximal subgroup of the Lyons group.

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 Co₃, McL has the normalizer McL:2, which is maximal in Co₃.

McL has 2 classes of maximal subgroups isomorphic to the Mathieu group 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 x = (−3, 1²³) and y = (−4,-4,0²²)'. The triangle's edge x-y = (1, 5, 1²²) is type 3; it is fixed by a Co₃. This M₂₂ is the monomial, and a maximal, subgroup of a representation of McL.

Wilson (2009) (p. 207) shows that the subgroup McL is well-defined. In the Leech lattice, suppose a type 3 point v is fixed by an instance of $\mathrm {Co} _{3}$ . 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 552 = 2³⋅3⋅23 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

Finkelstein (1973) 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. [1] The names of conjugacy classes are taken from the Atlas of Finite Group Representations.[2]

Cycle structures in the rank 3 permutation representation, degree 275, of McL are shown.[3]

Class	Centraliser order	No. elements	Trace	Cycle type
1A	898,128,000	1	24
2A	40,320	3⁴ ⋅ 5² ⋅ 11	8	1³⁵, 2¹²⁰
3A	29,160	2⁴ ⋅ 5² ⋅ 7 ⋅ 11	-3	1⁵, 3⁹⁰
3B	972	2³ ⋅ 3 ⋅ 5³ ⋅ 7 ⋅ 11	6	1¹⁴, 3⁸⁷
4A	96	2² ⋅ 3⁵ ⋅ 5³ ⋅ 7 ⋅ 11	4	1⁷, 2¹⁴, 4⁶⁰
5A	750	2⁶ ⋅ 3⁵ ⋅ ⋅ 7 ⋅ 11	-1	5⁵⁵
5B	25	2⁷ ⋅ 3⁶ ⋅ 5 ⋅ 7 ⋅ 11	4	1⁵, 5⁵⁴
6A	360	2⁴ ⋅ 3⁴ ⋅ 5² ⋅ 7 ⋅ 11	5	1⁵, 3¹⁰, 6⁴⁰
6B	36	2⁵ ⋅ 3⁴ ⋅ 5³ ⋅ 7 ⋅ 11	2	1², 2⁶, 3¹¹, 6³⁸
7A	14	2⁶ ⋅ 3⁶ ⋅ 5³ ⋅ 11	3	1², 7³⁹	power equivalent
7B	14	2⁶ ⋅ 3⁶ ⋅ 5³ ⋅ 11	3	1², 7³⁹	power equivalent
8A	8	2⁴ ⋅ 3⁶ ⋅ 5³ ⋅ 7 ⋅ 11	2	1, 2³, 4⁷, 8³⁰
9A	27	2⁷ ⋅ 3³ ⋅ 5³ ⋅ 7 ⋅ 11	3	1², 3, 9³⁰	power equivalent
9B	27	2⁷ ⋅ 3³ ⋅ 5³ ⋅ 7 ⋅ 11	3	1², 3, 9³⁰	power equivalent
10A	10	2⁶ ⋅ 3⁵ ⋅ 5³ ⋅ 7 ⋅ 11	3	5⁷, 10²⁴
11A	11	2⁷ ⋅ 3⁶ ⋅ 5³ ⋅ 7	2	11²⁵	power equivalent
11B	11	2⁷ ⋅ 3⁶ ⋅ 5³ ⋅ 7	2	11²⁵	power equivalent
12A	12	2⁵ ⋅ 3⁵ ⋅ 5³ ⋅ 7 ⋅ 11	1	1, 2², 3², 6⁴, 12²⁰
14A	14	2⁶ ⋅ 3⁶ ⋅ 5³ ⋅ 11	1	2, 7⁵, 14¹⁷	power equivalent
14B	14	2⁶ ⋅ 3⁶ ⋅ 5³ ⋅ 11	1	2, 7⁵, 14¹⁷	power equivalent
15A	30	2⁶ ⋅ 3⁵ ⋅ 5² ⋅ 7 ⋅ 11	2	5, 15¹⁸	power equivalent
15B	30	2⁶ ⋅ 3⁵ ⋅ 5² ⋅ 7 ⋅ 11	2	5, 15¹⁸	power equivalent
30A	30	2⁶ ⋅ 3⁵ ⋅ 5² ⋅ 7 ⋅ 11	0	5, 15², 30⁸	power equivalent
30B	30	2⁶ ⋅ 3⁵ ⋅ 5² ⋅ 7 ⋅ 11	0	5, 15², 30⁸	power equivalent

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 $T_{2A}(\tau )$ and $T_{4A}(\tau )$ .

References

Conway et al. (1985)
http://brauer.maths.qmul.ac.uk/Atlas/v3/permrep/McLG1-p275B0
http://brauer.maths.qmul.ac.uk/Atlas/v3/permrep/McLG1-p275B0

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.
Finkelstein, Larry (1973), "The maximal subgroups of Conway's group C₃ and McLaughlin's group", Journal of Algebra, 25: 58–89, doi:10.1016/0021-8693(73)90075-6, ISSN 0021-8693, MR 0346046
Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03516-0, ISBN 978-3-540-62778-4, MR 1707296
McLaughlin, Jack (1969), "A simple group of order 898,128,000", in Brauer, R.; Sah, Chih-han (eds.), Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), Benjamin, New York, pp. 109–111, MR 0242941
Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251, 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012

External links

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[1] Conway et al. (1985)

[2] ttp://brauer.maths.qmul.ac.uk/Atlas/v3/permrep/McLG1-p275B0

[3] ttp://brauer.maths.qmul.ac.uk/Atlas/v3/permrep/McLG1-p275B0