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 [math]\displaystyle{ \mathrm{Co}_0 }[/math], [math]\displaystyle{ \mathrm{Co}_2 }[/math], and [math]\displaystyle{ \mathrm{Co}_3 }[/math]. 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 [math]\displaystyle{ \mathrm{Co}_3 }[/math]. 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. 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.

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
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
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
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
15A	30	26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11	2	5, 1518	power equivalent
15B	30	26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11	2	5, 1518
30A	30	26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11	0	5, 152, 308	power equivalent
30B	30	26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11	0	5, 152, 308

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 [math]\displaystyle{ T_{2A}(\tau) }[/math] and [math]\displaystyle{ T_{4A}(\tau) }[/math].

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.
• 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 
• 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 
• McLaughlin, Jack (1969), "A simple group of order 898,128,000", in Brauer, R.; Sah, Chih-han, Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), Benjamin, New York, pp. 109–111 
• 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 

External links

• MathWorld: McLaughlin group
• Atlas of Finite Group Representations: McLaughlin group