Fischer group Fi22

Fi₂₂ is one of the 26 sporadic groups and is the smallest of the three Fischer groups. It was introduced by {{harvs|txt |first=Bernd |last=Fischer |authorlink=Bernd Fischer (mathematician) |year1=1971 |year2=1976}} while investigating 3-transposition groups.

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

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. {{harvtxt|Hiss|White|1994}} found the 5-modular character table, and {{harvtxt|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.

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 $T\_\{6A\}(\backslash tau)$ where one can set a(0) = 10 ({{OEIS2C|id=A007254}}),

:$\backslash begin\{align\}j\_\{6A\}(\backslash tau)$

&=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{align}

and η(τ) is the Dedekind eta function.

• {{Citation | last1=Aschbacher | first1=Michael | author1-link=Michael Aschbacher | title=3-transposition groups | publisher=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 | access-date=2012-06-21 | archive-date=2016-03-04 | archive-url=https://web.archive.org/web/20160304045542/http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511759413 | url-status=dead | url-access=subscription }} contains a complete proof of Fischer's theorem.
• {{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | editor1-last=Shult | editor1-first=and Ernest E. | editor2-last=Hale | editor2-first=Mark P. | editor3-last=Gagen | editor3-first=Terrence | title=Finite groups '72 (Proceedings of the Gainesville Conference on Finite Groups, University of Florida, Gainesville, Fla., March 23–24, 1972.) | publisher=North-Holland | location=Amsterdam | series=North-Holland Mathematics Studies | mr=0372016 | year=1973 | volume=7 | chapter=A construction for the smallest Fischer group F₂₂ | pages=27–35}}
• {{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=White | first2=Donald L. | last1=Hiss | first1=Gerhard | title=The 5-modular characters of the covering group of the sporadic simple Fischer group Fi₂₂ and its automorphism group | doi=10.1080/00927879408825043 | mr=1278807 | year=1994 | journal=Communications in Algebra | issn=0092-7872 | volume=22 | issue=9 | pages=3591–3611}}
• {{Citation | last1=Noeske | first1=Felix | title=The 2- and 3-modular characters of the sporadic simple Fischer group Fi₂₂ and its cover | doi=10.1016/j.jalgebra.2006.06.020 | mr=2303203 | year=2007 | journal=Journal of Algebra | issn=0021-8693 | volume=309 | issue=2 | pages=723–743| doi-access=free }}
• {{Citation | last1=Wilson | first1=Robert A. | title=On maximal subgroups of the Fischer group Fi₂₂ | doi=10.1017/S0305004100061491 | mr=735364 | year=1984 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | issn=0305-0041 | volume=95 | issue=2 | pages=197–222}}
• {{Citation | last1=Wilson | first1=Robert A. | title=The finite simple groups | publisher=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/F22/ ATLAS of Finite Group Representations.]

• [http://mathworld.wolfram.com/FischerGroups.html MathWorld: Fischer Groups]
• [http://brauer.maths.qmul.ac.uk/Atlas/v3/lookup?target=Fi22&SUBMIT=Go Atlas of Finite Group Representations: Fi22]

{{DEFAULTSORT:Fischer group Fi22}}

Category:Sporadic groups