Fischer group Fi23
|Algebraic structure → Group theory
In mathematics, the Fischer group Fi23 or M(23) or F23 of order
- 218 · 313 · 52 · 7 · 11 · 13 · 17 · 23 (= 4089470473293004800) is one of the three Fischer groups, sporadic simple groups introduced by Bernd Fischer (1971, 1976) while investigating 3-transposition groups.
The outer automorphism group has order 1, and the Schur multiplier has order 1.
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
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 where one can set the constant term a(0) = 42 ( A030197),
and so on. Like the j-function, the function j3A(τ) can also assume an integer value for appropriate arguments. For example,
Kleidman, Parker & Wilson (1989) found the conjugacy classes of maximal subgroups of Fi23 as follows:
O7(3) × S3
.(L3(3) × 2)
(22 × 21+8).(3 × U4(2)).2
26+8:(A7 × S3)
S6(2) × S4
- Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics 124, Cambridge University Press, doi:10.1017/CBO9780511759413, ISBN 978-0-521-57196-8, MR 1423599 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 The remainder of the paper is unpublished (as of 2010).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 989922
- Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792
- Wilson, R. A. ATLAS of Finite Group Representations.