View source for Fischer group Fi23

{{DISPLAYTITLE:Fischer group Fi<sub>23</sub>}}   
{{Group theory sidebar |Finite}}

In the area of modern algebra known as [[Group theory|group theory]], the '''Fischer group''' ''Fi<sub>23</sub>'' is a sporadic simple group of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;2<sup>18</sup>{{·}}3<sup>13</sup>{{·}}5<sup>2</sup>{{·}}7{{·}}11{{·}}13{{·}}17{{·}}23
: = 4089470473293004800
: ≈ 4{{e|18}}.

==History==
''Fi<sub>23</sub>'' is one of the 26 sporadic groups and is one of the three [[Fischer group]]s introduced by {{harvs|txt |first=Bernd |last=Fischer |year1=1971 |year2=1976}} while investigating [[3-transposition group|3-transposition group]]s.

The [[Schur multiplier]] and the [[Outer automorphism group|outer automorphism group]] are both [[Trivial group|trivial]].

==Representations==

The Fischer group Fi<sub>23</sub> 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

The smallest faithful complex representation has dimension <math>782</math>. 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|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 ''Fi''<sub>23</sub>, the relevant McKay-Thompson series is <math>T_{3A}(\tau)</math> where one can set the constant term a(0) = 42 ({{OEIS2C|id=A030197}}),

:<math>\begin{align}j_{3A}(\tau)
&=T_{3A}(\tau)+42\\
&=\left(\left(\tfrac{\eta(\tau)}{\eta(3\tau)}\right)^{6}+3^3 \left(\tfrac{\eta(2\tau)}{\eta(\tau)}\right)^{6}\right)^2\\
&=\frac{1}{q} + 42 + 783q + 8672q^2 +65367q^3+371520q^4+\dots
\end{align}</math>

and ''η''(''τ'') is the [[Dedekind eta function]].

== Maximal subgroups ==
{{harvtxt|Kleidman|Parker|Wilson|1989}} found the 14 conjugacy classes of maximal subgroups of ''Fi<sub>23</sub>'' as follows:

* 2.Fi<SUB>22</SUB>
* O<SUB>8</SUB><SUP>+</SUP>(3):S<SUB>3</SUB>
* 2<SUP>2</SUP>.U<SUB>6</SUB>(2).2
* S<SUB>8</SUB>(2)
* O<SUB>7</SUB>(3) × S<SUB>3</SUB>
* 2<SUP>11</SUP>.M<SUB>23</SUB>
* 3<SUP>1+8</SUP>.2<SUP>1+6</SUP>.3<SUP>1+2</SUP>.2S<SUB>4</SUB>
* [3<SUP>10</SUP>].(L<SUB>3</SUB>(3) × 2)
* S<SUB>12</SUB>
* (2<SUP>2</SUP> × 2<SUP>1+8</SUP>).(3 × U<SUB>4</SUB>(2)).2
* 2<SUP>6+8</SUP>:(A<SUB>7</SUB> × S<SUB>3</SUB>)
* S<SUB>6</SUB>(2) × S<SUB>4</SUB>
* S<SUB>4</SUB>(4):4
* L<SUB>2</SUB>(23)

== References ==

*{{Citation | last1=Aschbacher | first1=Michael |  title=3-transposition groups | publisher={{wipe|Cambridge University Press}} | series=Cambridge Tracts in Mathematics | isbn=978-0-521-57196-8 | year=1997 | volume=124 |url=http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511759413 | mr=1423599 | doi=10.1017/CBO9780511759413}} contains a complete proof of Fischer's theorem.
*{{Citation | last1=Fischer | first1=Bernd | title=Finite groups generated by 3-transpositions. I | doi=10.1007/BF01404633 | year=1971 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=13 | pages=232–246 | mr=0294487 | issue=3}} 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).
*{{Citation | last1=Fischer | first1=Bernd | title=Finite Groups Generated by 3-transpositions | url=https://books.google.com/books?id=PjezNwAACAAJ | publisher=Mathematics Institute, University of Warwick | series=Preprint | year=1976}}
*{{Citation | last2=Parker | first2=Richard A. | last1=Kleidman | first1=Peter B. | last3=Wilson | first3=Robert A. | title=The maximal subgroups of the Fischer group Fi₂₃ | doi=10.1112/jlms/s2-39.1.89 | mr=989922  | year=1989 | journal=Journal of the London Mathematical Society |series=Second Series | issn=0024-6107 | volume=39 | issue=1 | pages=89–101}}
* {{Citation | last1=Wilson | first1=Robert A. | title=The finite simple groups | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics 251 | isbn=978-1-84800-987-5 | doi=10.1007/978-1-84800-988-2 | year=2009 | zbl=1203.20012 | volume=251}}
*Wilson, R. A.  [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/F23/ ATLAS of Finite Group Representations.]

== External links ==
* [http://mathworld.wolfram.com/FischerGroups.html MathWorld: Fischer Groups]
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/lookup?target=Fi23&SUBMIT=Go Atlas of Finite Group Representations: Fi23]

{{DEFAULTSORT:Fischer group Fi23}}
[[Category:Sporadic groups]]

{{Sourceattribution|Fischer group Fi23}}