Fischer Group - 3-transposition Groups

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 become 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.

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, 8 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.