Rudvalis group

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

   2¹⁴ · 3³ · 5³ · 7 · 13 · 29

= 145926144000

≈ 1×10¹¹.

History

Ru is one of the 26 sporadic groups and was found by Arunas Rudvalis (1973, 1984) and constructed by Conway and Wales (1973). Its Schur multiplier has order 2, and its outer automorphism group is trivial.

In 1982 R. L. Griess showed that Ru cannot be a subquotient of the monster group.^[1] Thus it is one of the 6 sporadic groups called the pariahs.

Properties

The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer the Ree group ²F₄(2), the automorphism group of the Tits group. This representation implies a strongly regular graph in which each vertex has 2304 neighbors and 1755 non-neighbors. Two adjacent vertices have 1328 common neighbors; two non-adjacent ones have 1208 (Griess 1998, p. 125).

Its double cover acts on a 28-dimensional lattice over the Gaussian integers. The lattice has 4×4060 minimal vectors; if minimal vectors are identified if one is 1, i, –1, or –i times another then the 4060 equivalence classes can be identified with the points of the rank 3 permutation representation. Reducing this lattice modulo the principal ideal

gives an action of the Rudvalis group on a 28-dimensional vector space over the field with 2 elements. Duncan (2006) used the 28-dimensional lattice to construct a vertex operator algebra acted on by the double cover.

Parrott (1976) characterized the Rudvalis group by the centralizer of a central involution. Aschbacher & Smith (2004) gave another characterization as part of their identification of the Rudvalis group as one of the quasithin groups.

Maximal subgroups

Wilson (1984) found the 15 conjugacy classes of maximal subgroups of Ru as follows:

• ²F₄(2) = ²F₄(2)'.2
• 2⁶.U₃(3).2
• (2² × Sz(8)):3
• 2³⁺⁸:L₃(2)
• U₃(5):2
• 2¹⁺⁴⁺⁶.S₅
• PSL₂(25).2²
• A₈
• PSL₂(29)
• 5²:4.S₅
• 3.A₆.2²
• 5¹⁺²:[2⁵]
• L₂(13):2
• A₆.2²
• 5:4 × A₅

References

• Aschbacher, Michael; Smith, Stephen D. (2004), The classification of quasithin groups. I Structure of Strongly Quasithin K-groups, Mathematical Surveys and Monographs, 111, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3410-7, MR 2097623 
• Conway, J.H.; Wales, D.B. (1973), "The construction of the Rudvalis simple group of order 145926144000", Journal of Algebra, 27 (27): 538–548, doi:10.1016/0021-8693(73)90063-X 
• John F. Duncan (2008). "Moonshine for Rudvalis's sporadic group". arXiv:math/0609449v1 [math.RT]. 
• Griess, R.L. (1982), "The Friendly Giant", Inventiones Mathematicae, 69 (69): 1–102, doi:10.1007/BF01389186 
• Griess, R.L. (1998), Twelve Sporadic Groups, Springer-Verlag 
• Parrott, David (1976), "A characterization of the Rudvalis simple group", Proceedings of the London Mathematical Society. Third Series, 32 (1): 25–51, doi:10.1112/plms/s3-32.1.25, ISSN 0024-6115, MR 0390043 
• Rudvalis, A. (1973), "A new simple group of order 2¹⁴ 3³ 5³ 7 13 29", Notices of the American Mathematical Society (20): A–95 
• Rudvalis, Arunas (1984), "A rank 3 simple group of order 2¹⁴3³5³7.13.29. I", Journal of Algebra, 86 (1): 181–218, doi:10.1016/0021-8693(84)90063-2, ISSN 0021-8693, MR 727376 
• Rudvalis, Arunas (1984), "A rank 3 simple group G of order 2¹⁴3³5³7.13.29. II. Characters of G and Ĝ", Journal of Algebra, 86 (1): 219–258, doi:10.1016/0021-8693(84)90064-4, ISSN 0021-8693, MR 727377 
• Wilson, Robert A. (1984), "The geometry and maximal subgroups of the simple groups of A. Rudvalis and J. Tits", Proceedings of the London Mathematical Society. Third Series, 48 (3): 533–563, doi:10.1112/plms/s3-48.3.533, ISSN 0024-6115, MR 735227 

External links

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