Fischer group Fi23

Last updated October 25, 2024

In the area of modern algebra known as group theory, the Fischer groupFi₂₃ is a sporadic simple group of order

History

Fi₂₃ is one of the 26 sporadic groups and is one of the three Fischer groups introduced by BerndFischer ( 1971 , 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 $782$ . 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 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 Fi₂₃, the relevant McKay-Thompson series is $T_{3A}(\tau )$ where one can set the constant term a(0) = 42 ( OEIS: A030197 ),

{\begin{aligned}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{aligned}}

and η(τ) is the Dedekind eta function.

Maximal subgroups

Kleidman, Parker & Wilson (1989) found the 14 conjugacy classes of maximal subgroups of Fi₂₃ as follows:

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

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References

Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics, vol. 124, Cambridge University Press, doi:10.1017/CBO9780511759413, ISBN 978-0-521-57196-8, MR 1423599, archived from the original on 2016-03-04, retrieved 2012-06-21 contains a complete proof of Fischer's theorem.
Fischer, Bernd (1971), "Finite groups generated by 3-transpositions. I", Inventiones Mathematicae , 13 (3): 232–246, doi:10.1007/BF01404633, ISSN 0020-9910, MR 0294487 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).
Fischer, Bernd (1976), Finite Groups Generated by 3-transpositions, Preprint, Mathematics Institute, University of Warwick
Kleidman, Peter B.; Parker, Richard A.; Wilson, Robert A. (1989), "The maximal subgroups of the Fischer group Fi₂₃", Journal of the London Mathematical Society, Second Series, 39 (1): 89–101, doi:10.1112/jlms/s2-39.1.89, ISSN 0024-6107, MR 0989922
Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012
Wilson, R. A. ATLAS of Finite Group Representations.

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