Fischer group Fi23

{{Short description|Sporadic simple group}} {{for|general background and history of the Fischer sporadic groups|Fischer group}} {{DISPLAYTITLE:Fischer group Fi₂₃}} {{Group theory sidebar |Finite}}

In the area of modern algebra known as group theory, the '''Fischer group''' ''Fi₂₃'' is a sporadic simple group of order :   4,089,470,473,293,004,800 : = 2¹⁸{{·}}3¹³{{·}}5²{{·}}7{{·}}11{{·}}13{{·}}17{{·}}23 : ≈ 4{{e|18}}.

==History== ''Fi₂₃'' is one of the 26 sporadic groups and is one of the three Fischer groups introduced by {{harvs|txt |first=Bernd |last=Fischer |authorlink=Bernd Fischer (mathematician) |year1=1971 |year2=1976}} while investigating 3-transposition groups.

The Schur multiplier and the outer automorphism group are both trivial.

==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. It has a second rank-3 action on 137632 points

Fi₂₃ is the centralizer of a transposition in the Fischer group Fi24. When realizing Fi₂₄ as a subgroup of the Monster group, the full centralizer of a transposition is the double cover of the Baby monster group. As a result, Fi₂₃ is a subgroup of the Baby monster, and is the normalizer of a certain S₃ group in the Monster.

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 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 <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 ''Fi₂₃'' as follows:

{| class="wikitable" |+ Maximal subgroups of ''Fi₂₃'' |- ! No. !! Structure !! Order !! Index !! Comments |- | 1||2.Fi₂₂ ||align=right|129,123,503,308,800 <br />=&nbsp; 2¹⁸·3⁹·5²·7·11·13||align=right| 31,671<br />=&nbsp;3⁴·17·23 || centralizer of an involution of class 2A |- | 2||O{{su|a=c|b=8|p=+}}(3):S₃ ||align=right| 29,713,078,886,400 <br />=&nbsp; 2¹³·3¹³·5²·7·13 ||align=right| 137,632<br />=&nbsp;2⁵·11·17·23 || |- | 3||2².U₆(2).2 ||align=right| 73,574,645,760 <br />=&nbsp; 2¹⁸·3⁶·5·7·11 ||align=right| 55,582,605<br />=&nbsp;3⁷·5·13·17·23 || centralizer of an involution of class 2B |- | 4||S₈(2) ||align=right| 47,377,612,800 <br />=&nbsp; 2¹⁶·3⁵·5²·7·17 ||align=right| 86,316,516<br />=&nbsp;2²·3⁸·11·13·23 || |- | 5||O₇(3) × S₃ ||align=right| 27,512,110,080 <br />=&nbsp; 2¹⁰·3¹⁰·5·7·13 ||align=right| 148,642,560<br />=&nbsp;2⁸·3³·5·11·17·23 || normalizer of a subgroup of order 3 (class 3A) |- | 6||2¹¹.M₂₃ ||align=right| 20,891,566,080 <br />=&nbsp; 2¹⁸·3²·5·7·11·23 ||align=right| 195,747,435<br />=&nbsp;3¹¹·5·13·17 || |- | 7||3¹⁺⁸.2¹⁺⁶.3¹⁺².2S₄||align=right| 3,265,173,504 <br />=&nbsp; 2¹¹·3¹³ ||align=right| 1,252,451,200<br />=&nbsp;2⁷·5²·7·11·13·17·23 || normalizer of a subgroup of order 3 (class 3B) |- | 8||[3¹⁰].(L₃(3) × 2) ||align=right| 663,238,368 <br />=&nbsp; 2⁵·3¹³·13 ||align=right| 6,165,913,600<br />=&nbsp;2¹³·5²·7·11·17·23 || |- | 9||S₁₂ ||align=right| 479,001,600 <br />=&nbsp; 2¹⁰·3⁵·5²·7·11 ||align=right| 8,537,488,128<br />=&nbsp;2⁸·3⁸·13·17·23 || |- |10||(2² × 2¹⁺⁸).(3 × U₄(2)).2 ||align=right| 318,504,960 <br />=&nbsp; 2¹⁸·3⁵·5 ||align=right| 12,839,581,755<br />=&nbsp;3⁸·5·7·11·13·17·23 || centralizer of an involution of class 2C |- |11||2⁶⁺⁸:(A₇ × S₃) ||align=right| 247,726,080 <br />=&nbsp; 2¹⁸·3³·5·7 ||align=right| 16,508,033,685<br />=&nbsp;3¹⁰·5·11·13·17·23 || |- |12||S₆(2) × S₄ ||align=right| 34,836,480 <br />=&nbsp; 2¹²·3⁵·5·7 ||align=right| 117,390,461,760<br />=&nbsp;2⁶·3⁸·5·11·13·17·23 || |- |13||S₄(4):4 ||align=right| 3,916,800 <br />=&nbsp; 2¹⁰·3²·5²·17 ||align=right| 1,044,084,577,536<br />=&nbsp;2⁸·3¹¹·7·11·13·23 || |- |14||L₂(23) ||align=right| 6,072 <br />=&nbsp; 2³·3·11·23 ||align=right|673,496,454,758,400<br />=&nbsp;2¹⁵·3¹²·5²·7·13·17|| |}

== References ==

*{{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=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=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