Fischer groups

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).

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

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 2364, 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.

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