| Algebraic structure → Group theory Group theory |
|---|
 |
In the area of modern algebra known as group theory, the Rudvalis groupRu is a sporadic simple group of order
Contents
- 214 ·33 ·53 ·7 ·13 ·29
- = 145926144000
- ≈ 1×1011.
| Algebraic structure → Group theory Group theory |
|---|
| |
In the area of modern algebra known as group theory, the Rudvalis groupRu is a sporadic simple group of order
Ru is one of the 26 sporadic groups and was found by ArunasRudvalis ( 1973 , 1984 ) and constructed by John H.Conway andDavid 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. [1] Thus it is one of the 6 sporadic groups called the pariahs.
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
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.
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.
Wilson (1984) found the 15 conjugacy classes of maximal subgroups of Ru as follows:
In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.
In the mathematical classification of finite simple groups, there are 26 or 27 groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups.
In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by (Conway 1968, 1969).
In group theory, the Tits group2F4(2)′, named for Jacques Tits (French:[tits]), is a finite simple group of order
In the area of modern algebra known as group theory, the Thompson groupTh is a sporadic simple group of order
In the area of abstract algebra known as group theory, the O'Nan groupO'N or O'Nan–Sims group is a sporadic simple group of order
In mathematics, a Ree group is a group of Lie type over a finite field constructed by Ree from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a different method. They were the last of the infinite families of finite simple groups to be discovered.
In the area of modern algebra known as group theory, the Janko groupJ1 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Janko groupJ4 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Janko groupJ2 or the Hall-Janko groupHJ is a sporadic simple group of order
In the area of modern algebra known as group theory, the Mathieu groupM12 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Mathieu groupM22 is a sporadic simple group of order
In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order
In finite group theory, an area of abstract algebra, a strongly embedded subgroup of a finite group G is a proper subgroup H of even order such that H ∩ Hg has odd order whenever g is not in H. The Bender–Suzuki theorem, proved by Bender (1971) extending work of Suzuki (1962, 1964), classifies the groups G with a strongly embedded subgroup H. It states that either
In the mathematical classification of finite simple groups, a thin group is a finite group such that for every odd prime number p, the Sylow p-subgroups of the 2-local subgroups are cyclic. Informally, these are the groups that resemble rank 1 groups of Lie type over a finite field of characteristic 2.
In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 215·32·5·7·31, that is the unique nonsplit extension of by its natural module of order . The uniqueness of such a nonsplit extension was shown by Dempwolff (1972), and the existence by Thompson (1976), who showed using some computer calculations of Smith (1976) that the Dempwolff group is contained in the compact Lie group as the subgroup fixing a certain lattice in the Lie algebra of , and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.
In the area of modern algebra known as group theory, the Fischer groupFi22 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Conway groupCo2 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Conway groupCo1 is a sporadic simple group of order
