Rudvalis group
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In the mathematical field of group theory, the Rudvalis group Ru (found by Arunas Rudvalis (1973) and constructed by Conway and Wales (1973)) is a sporadic simple group of order
- 214 · 33 · 53 · 7 · 13 · 29
- = 145926144000
- ≈ 1011.
Ru is one of the six sporadic simple groups called the pariahs, because they are not found within the Monster group (Griess 1983, p. 91).
[edit] Properties
The Rudvalis group act 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 1255 non-neighbors. Two adjacent vertices have 1328 common neighbors; two non-adjacent ones have 1208 (Griess 1998, p. 125)
Its Schur multiplier has order 2, and its outer automorphism group is trivial.
Its double cover acts on a 28-dimensional lattice over the Gaussian integers. 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 with 2 elements. Duncan (2006) used this to construct a vertex operator algebra acted on by the double cover.
A novel simple subgroup of Ru is the Suzuki group Sz(8), order 21920, which is not divisible by 3. (Atlas of Finite Group Representations: Rudvalis group)
[edit] References
- Conway, J.H. & Wales, D.B. (1973), “The construction of the Rudvalis simple group of order 145926144000”, Journal of Algebra (no. 27): 538-548
- John F. Duncan (2008). "Moonshine for Rudvalis's sporadic group" arxiv:math/0609449v1 [math.RT].
- Griess, R.L. (1982), “The Friendly Giant”, Inventiones Mathematicae (no. 69): 1-102
- Griess, R.L. (1998), Twelve Sporadic Groups, Springer-Verlag
- Rudvalis, A. (1973), “A new simple group of order 214 33 53 7 13 29”, Notices of the American Mathematical Society (no. 20): A-95
