| Algebraic structure → Group theory Group theory |
|---|
 |
In the area of modern algebra known as group theory, the Fischer groupFi23 is a sporadic simple group of order
Contents
- 218 ·313 ·52 ·7 ·11 ·13 ·17 ·23
- = 4089470473293004800
- ≈ 4×1018.
| Algebraic structure → Group theory Group theory |
|---|
| |
In the area of modern algebra known as group theory, the Fischer groupFi23 is a sporadic simple group of order
Fi23 is one of the 26 sporadic groups and is one of the three Fischer groups introduced by BerndFischer ( 1971 , 1976 ) while investigating 3-transposition groups.
The Schur multiplier and the outer automorphism group are both trivial.
The Fischer group Fi23 has a rank 3 action on a graph of 31671 vertices corresponding to 3-transpositions, with point stabilizer the double cover of the Fischer group Fi22. It has a second rank-3 action on 137632 points
Fi23 is the centralizer of a transposition in the Fischer group Fi24. When realizing Fi24 as a subgroup of the Monster group, the full centralizer of a transposition is the double cover of the Baby monster group. As a result, Fi23 is a subgroup of the Baby monster, and is the normalizer of a certain S3 group in the Monster.
The smallest faithful complex representation has dimension . The group has an irreducible representation of dimension 253 over the field with 3 elements.
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For Fi23, the relevant McKay-Thompson series is where one can set the constant term a(0) = 42 ( OEIS: A030197 ),
and η(τ) is the Dedekind eta function.
Kleidman, Parker & Wilson (1989) found the 14 conjugacy classes of maximal subgroups of Fi23 as follows:
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem.
In the area of modern algebra known as group theory, the baby monster groupB (or, more simply, the baby monster) is a sporadic simple group of order
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 Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by Bernd Fischer (1971, 1976).
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 the area of modern algebra known as group theory, the Higman–Sims group HS is a sporadic simple group of order
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory.
In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL(2, Z) defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that
In the area of modern algebra known as group theory, the Thompson groupTh is a sporadic simple group of order
In the area of modern algebra known as group theory, the Held groupHe is a sporadic simple group of order
In the area of modern algebra known as group theory, the Harada–Norton groupHN is a sporadic simple group of order
Robert Arnott Wilson is a retired mathematician in London, England, who is best known for his work on classifying the maximal subgroups of finite simple groups and for the work in the Monster group. He is also an accomplished violin, viola and piano player, having played as the principal viola in the Sinfonia of Birmingham. Due to a damaged finger, he now principally plays the kora.
In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14. The explanation for this phenomenon is arithmetic. Namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the triplet of Riemann surfaces.
In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order
The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry, namely the Hurwitz surfaces. The Hurwitz quaternion order was studied in 1967 by Goro Shimura, but first explicitly described by Noam Elkies in 1998. For an alternative use of the term, see Hurwitz quaternion.
In mathematical group theory, a 3-transposition group is a group generated by a conjugacy class of involutions, called the 3-transpositions, such that the product of any two involutions from the conjugacy class has order at most 3.
In mathematics, the Weber modular functions are a family of three functions f, f1, and f2, studied by Heinrich Martin Weber.
In the area of modern algebra known as group theory, the Fischer groupFi24 or F24′ or F3+ is a sporadic simple group of order
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 group is a sporadic simple group of order
