Fischer Group Fi22
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In the area of modern algebra known as
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the Fischer group ''Fi22'' is a
sporadic simple group In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...
of order :   217395271113 : = 64561751654400 : ≈ 6.


History

''Fi22'' is one of the 26 sporadic groups and is the smallest of the three Fischer groups. It was introduced by while investigating 3-transposition groups. The
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
has order 2, and the
Schur multiplier In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \oper ...
has order 6.


Representations

The Fischer group Fi22 has a
rank 3 action Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
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. found the 5-modular character table, and found the 2- and 3-modular character tables. The automorphism group of Fi22 centralizes an element of order 3 in the
baby monster group In the area of modern algebra known as group theory, the baby monster group ''B'' (or, more simply, the baby monster) is a sporadic simple group of order :   241313567211131719233147 : = 4154781481226426191177580544000000 : = 4 ...
.


Generalized Monstrous Moonshine

Conway and Norton suggested in their 1979 paper that
monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
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 ''Fi''22, the McKay-Thompson series is T_(\tau) where one can set a(0) = 10 (), :\beginj_(\tau) &=T_(\tau)+10\\ &=\left(\left(\tfrac\right)^+2^3 \left(\tfrac\right)^\right)^2\\ &=\left(\left(\tfrac\right)^+3^2 \left(\tfrac\right)^\right)^2-4\\ &=\frac + 10 + 79q + 352q^2 +1431q^3+4160q^4+13015q^5+\dots \end and ''η''(''τ'') is the
Dedekind eta function 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 t ...
.


Maximal subgroups

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(2):S3 * 210:M22 * 26:S6(2) * (2 × 21+8):(U4(2):2) * U4(3):2 × S3 * 2F4(2)' (This is the
Tits group In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order :   211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a 27th sporadic group. ...
) * 25+8:(S3 × A6) * 31+6:23+4:32:2 * S10 (Two classes, fused by an outer automorphism) * M12


References

* contains a complete proof of Fischer's theorem. * * 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). * * * * * *Wilson, R. A.
ATLAS of Finite Group Representations.


External links




Atlas of Finite Group Representations: Fi22
Sporadic groups {{Improve categories, date=August 2021