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 ''Fi₂₃'' 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 : 2¹⁸3¹³5²711131723 : = 4089470473293004800 : ≈ 4.

History

''Fi₂₃'' is one of the 26 sporadic groups and is one of the three

Fischer group 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 . 3-transposition groups The Fischer groups are named after Bernd Fischer who discovered them ...

s introduced by while investigating 3-transposition groups. 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 ...

and 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 ...

are both

trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...

Representations

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 In the area of modern algebra known as group theory, the Fischer group ''Fi22'' is a sporadic simple group of order : 217395271113 : = 64561751654400 : ≈ 6. History ''Fi22'' is one of the 26 sporadic groups and is the sma ...

. It has a second rank-3 action on 137632 points The smallest faithful complex representation has dimension

782

. The group has an irreducible representation of dimension 253 over the field with 3 elements.

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''₂₃, the relevant McKay-Thompson series is

T_(\tau)

where one can set the constant term a(0) = 42 (), :

\beginj_(\tau)
&=T_(\tau)+42\\
&=\left(\left(\tfrac\right)^+3^3 \left(\tfrac\right)^\right)^2\\
&=\frac + 42 + 783q + 8672q^2 +65367q^3+371520q^4+\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 14 conjugacy classes of maximal subgroups of ''Fi₂₃'' as follows: * 2.Fi₂₂ * O₈⁺(3):S₃ * 2².U₆(2).2 * S₈(2) * O₇(3) × S₃ * 2¹¹.M₂₃ * 3¹⁺⁸.2¹⁺⁶.3¹⁺².2S₄ * ¹⁰(L₃(3) × 2) * S₁₂ * (2² × 2¹⁺⁸).(3 × U₄(2)).2 * 2⁶⁺⁸:(A₇ × S₃) * S₆(2) × S₄ * S₄(4):4 * L₂(23)

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: Fi23
{{DEFAULTSORT:Fischer group Fi23 Sporadic groups