Fischer group

In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi₂₂, Fi₂₃ and Fi₂₄ introduced by Bernd Fischer (1971, 1976).

3-transposition groups

The Fischer groups are named after Bernd Fischer who discovered them while investigating 3-transposition groups. These are groups G with the following properties:

G is generated by a conjugacy class of elements of order 2, called 'Fischer transpositions' or 3-transpositions.
The product of any two distinct transpositions has order 2 or 3.

The typical example of a 3-transposition group is a symmetric group, where the Fischer transpositions are genuinely transpositions. The symmetric group S_n can be generated by n − 1 transpositions: (12), (23), ..., (n − 1, n).

Fischer was able to classify 3-transposition groups that satisfy certain extra technical conditions. The groups he found fell mostly into several infinite classes (besides symmetric groups: certain classes of symplectic, unitary, and orthogonal groups), but he also found 3 very large new groups. These groups are usually referred to as Fi₂₂, Fi₂₃ and Fi₂₄. The first two of these are simple groups, and the third contains the simple group Fi₂₄′ of index 2.

A starting point for the Fischer groups is the unitary group PSU₆(2), which could be thought of as a group Fi₂₁ in the series of Fischer groups, of order 9,196,830,720 = 2¹⁵⋅3⁶⋅5⋅7⋅11. Actually it is the double cover 2.PSU₆(2) that becomes a subgroup of the new group. This is the stabilizer of one vertex in a graph of 3510 (= 2⋅3³⋅5⋅13). These vertices are identified as conjugate 3-transpositions in the symmetry group Fi₂₂ of the graph.

The Fischer groups are named by analogy with the large Mathieu groups. In Fi₂₂ a maximal set of 3-transpositions all commuting with one another has size 22 and is called a basic set. There are 1024 3-transpositions, called anabasic that do not commute with any in the particular basic set. Any one of other 2464, called hexadic, commutes with 6 basic ones. The sets of 6 form an S(3,6,22) Steiner system, whose symmetry group is M₂₂. A basic set generates an abelian group of order 2¹⁰, which extends in Fi₂₂ to a subgroup 2¹⁰:M₂₂.

The next Fischer group comes by regarding 2.Fi₂₂ as a one-point stabilizer for a graph of 31671 (= 3⁴⋅17⋅23) vertices, and treating these vertices as the 3-transpositions in a group Fi₂₃. The 3-transpositions come in basic sets of 23, 7 of which commute with a given outside 3-transposition.

Next one takes Fi₂₃ and treats it as a one-point stabilizer for a graph of 306936 (= 2³⋅3³⋅7²⋅29) vertices to make a group Fi₂₄. The 3-transpositions come in basic sets of 24, eight of which commute with a given outside 3-transposition. The group Fi₂₄ is not simple, but its derived subgroup has index 2 and is a sporadic simple group.

Notation

There is no uniformly accepted notation for these groups. Some authors use F in place of Fi (F₂₂, for example). Fischer's notation for them was M(22), M(23) and M(24)′, which emphasised their close relationship with the three largest Mathieu groups, M₂₂, M₂₃ and M₂₄.

One particular source of confusion is that Fi₂₄ is sometimes used to refer to the simple group Fi₂₄′, and is sometimes used to refer to the full 3-transposition group (which is twice the size).

Generalized Monstrous Moonshine

John H. Conway and Simon P. 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 (main or principal moduli) from simple combinations of dimensions of sporadic groups.

References

Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics, vol. 124, Cambridge University Press, doi:10.1017/CBO9780511786313, ISBN 978-0-521-57196-8, MR 1423599, archived from the original on 2016-03-04, retrieved 2010-12-06 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
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 Representation"
https://web.archive.org/web/20171204142908/http://for.mat.bham.ac.uk/atlas/html/contents.html#spo