Rudvalis group

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 John H. Conway and David B. Wales (1973). Its Schur multiplier has order 2, and its outer automorphism group is trivial.

In 1982 Robert Griess showed that Ru cannot be a subquotient of the monster group. 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 being the Ree group ²F₄(2), the automorphism group of the Tits group. This representation implies a strongly regular graph srg(4060, 2304, 1328, 1280). That is, each vertex has 2304 neighbors and 1755 non-neighbors, any two adjacent vertices have 1328 common neighbors, while any two non-adjacent ones have 1280 (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 whenever 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

$(1+i)\$

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

Alternatively, the double cover can be defined abstractly, by starting with the graph and lifting Ru to 2Ru in the double cover 2A₄₀₆₀. This is because 1 of the conjugacy classes of involutions does not fix any points. Such an involution partitions the 4060 points of the graph into 2030 pairs, which can be regarded as 1015 double transpositions in the alternating group A₄₀₆₀. Since 1015 is odd, these involutions are lifted to order 4 elements in the double cover 2A₄₀₆₀. For more information, see Covering groups of the alternating and symmetric groups.

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, vol. 111, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3410-7, MR 2097623
• Conway, John H.; Wales, David B. (1973), "The construction of the Rudvalis simple group of order 145926144000", Journal of Algebra, 27 (3): 538–548, doi:10.1016/0021-8693(73)90063-X
• John F. Duncan (2008). "Moonshine for Rudvalis's sporadic group". arXiv:math/0609449v1.
• Griess, Robert L. (1982), "The Friendly Giant" (PDF), Inventiones Mathematicae, 69 (1): 1–102, Bibcode:1982InMat..69....1G, doi:10.1007/BF01389186, hdl:2027.42/46608
• Griess, Robert 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, Arunas (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 0727376
• 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 0727377
• 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 0735227

External links

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