Fischer group Fi22

Last updated July 26, 2022

In the area of modern algebra known as group theory, the Fischer groupFi₂₂ is a sporadic simple group of order

History

Fi₂₂ is one of the 26 sporadic groups and is the smallest of the three Fischer groups. It was introduced by BerndFischer ( 1971 , 1976 ) while investigating 3-transposition groups.

The outer automorphism group has order 2, and the Schur multiplier has order 6.

Representations

The Fischer group Fi₂₂ has a rank 3 action on a graph of 3510 vertices corresponding to its 3-transpositions, with point stabilizer the double cover of the group PSU₆(2). It also has two rank 3 actions on 14080 points, exchanged by an outer automorphism.

Fi₂₂ has an irreducible real representation of dimension 78. Reducing an integral form of this mod 3 gives a representation of Fi₂₂ over the field with 3 elements, whose quotient by the 1-dimensional space of fixed vectors is a 77-dimensional irreducible representation.

The perfect triple cover of Fi₂₂ has an irreducible representation of dimension 27 over the field with 4 elements. This arises from the fact that Fi₂₂ is a subgroup of ²E₆(2²). All the ordinary and modular character tables of Fi₂₂ have been computed. Hiss & White (1994) found the 5-modular character table, and Noeske (2007) found the 2- and 3-modular character tables.

The automorphism group of Fi₂₂ centralizes an element of order 3 in the baby monster group.

Generalized Monstrous Moonshine

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 Fi₂₂, the McKay-Thompson series is $T_{6A}(\tau )$ where one can set a(0) = 10 ( OEIS: A007254 ),

{\begin{aligned}j_{6A}(\tau )&=T_{6A}(\tau )+10\\&=\left(\left({\tfrac {\eta (\tau )\,\eta (3\tau )}{\eta (2\tau )\,\eta (6\tau )}}\right)^{3}+2^{3}\left({\tfrac {\eta (2\tau )\,\eta (6\tau )}{\eta (\tau )\,\eta (3\tau )}}\right)^{3}\right)^{2}\\&=\left(\left({\tfrac {\eta (\tau )\,\eta (2\tau )}{\eta (3\tau )\,\eta (6\tau )}}\right)^{2}+3^{2}\left({\tfrac {\eta (3\tau )\,\eta (6\tau )}{\eta (\tau )\,\eta (2\tau )}}\right)^{2}\right)^{2}-4\\&={\frac {1}{q}}+10+79q+352q^{2}+1431q^{3}+4160q^{4}+13015q^{5}+\dots \end{aligned}}

and η(τ) is the Dedekind eta function.

Maximal subgroups

Wilson (1984) found the 12 conjugacy classes of maximal subgroups of Fi₂₂ as follows:

2·U₆(2)
O₇(3) (Two classes, fused by an outer automorphism)
O⁺
₈(2):S₃
2¹⁰:M₂₂
2⁶:S₆(2)
(2 × 2¹⁺⁸):(U₄(2):2)
U₄(3):2 × S₃
²F₄(2)' (This is the Tits group)
2⁵⁺⁸:(S₃ × A₆)
3¹⁺⁶:2³⁺⁴:3²:2
S₁₀ (Two classes, fused by an outer automorphism)
M₁₂

Related Research Articles

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself ; in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.

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 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, theta functions are special functions of several complex variables. They are important in many areas, including the theories of Abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field 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 mathematics, E₈ is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E₈ comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled A_n, B_n, C_n, D_n, and five exceptional cases labeled G₂, F₄, E₆, E₇, and E₈. The E₈ algebra is the largest and most complicated of these exceptional cases.

In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers $. This problem, first posed in the early 19th century, is unsolved.$

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

Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions — often with desirable properties — that do not fit into any series. These are known as exceptional objects. In many cases, these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics often relate to the exceptional objects in others.

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

In mathematics, the Weber modular functions are a family of three functions f, f₁, and f₂, studied by Heinrich Martin Weber.

In mathematics, the Steinberg triality groups of type ³D₄ form a family of Steinberg or twisted Chevalley groups. They are quasi-split forms of D₄, depending on a cubic Galois extension of fields K ⊂ L, and using the triality automorphism of the Dynkin diagram D₄. Unfortunately the notation for the group is not standardized, as some authors write it as ³D₄(K) (thinking of ³D₄ as an algebraic group taking values in K) and some as ³D₄(L) (thinking of the group as a subgroup of D₄(L) fixed by an outer automorphism of order 3). The group ³D₄ is very similar to an orthogonal or spin group in dimension 8.

In the area of modern algebra known as group theory, the Fischer groupFi₂₄ or F₂₄′ is a sporadic simple group of order

In the area of modern algebra known as group theory, the Fischer groupFi₂₃ 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

In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as,

References

Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics, vol. 124, Cambridge University Press, doi:10.1017/CBO9780511759413, ISBN 978-0-521-57196-8, MR 1423599 contains a complete proof of Fischer's theorem.
Conway, John Horton (1973), "A construction for the smallest Fischer group F₂₂", in Shult, and Ernest E.; Hale, Mark P.; Gagen, Terrence (eds.), Finite groups '72 (Proceedings of the Gainesville Conference on Finite Groups, University of Florida, Gainesville, Fla., March 23–24, 1972.), North-Holland Mathematics Studies, vol. 7, Amsterdam: North-Holland, pp. 27–35, MR 0372016
Fischer, Bernd (1971), "Finite groups generated by 3-transpositions. I", Inventiones Mathematicae , 13 (3): 232–246, doi:10.1007/BF01404633, ISSN 0020-9910, MR 0294487 This is the first part of Fischer's preprint on the construction of his groups. The remainder of the paper is unpublished (as of 2010).
Fischer, Bernd (1976), Finite Groups Generated by 3-transpositions, Preprint, Mathematics Institute, University of Warwick
Hiss, Gerhard; White, Donald L. (1994), "The 5-modular characters of the covering group of the sporadic simple Fischer group Fi₂₂ and its automorphism group", Communications in Algebra, 22 (9): 3591–3611, doi:10.1080/00927879408825043, ISSN 0092-7872, MR 1278807
Noeske, Felix (2007), "The 2- and 3-modular characters of the sporadic simple Fischer group Fi₂₂ and its cover", Journal of Algebra , 309 (2): 723–743, doi: 10.1016/j.jalgebra.2006.06.020 , ISSN 0021-8693, MR 2303203
Wilson, Robert A. (1984), "On maximal subgroups of the Fischer group Fi₂₂", Mathematical Proceedings of the Cambridge Philosophical Society, 95 (2): 197–222, doi:10.1017/S0305004100061491, ISSN 0305-0041, MR 0735364
Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012
Wilson, R. A. ATLAS of Finite Group Representations.

External links

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.