Fischer (1964) first studied 3-transposition groups in the special case when the product of any two distinct 3-transpositions has order 3.
He showed that a finite group with this property is solvable, and has a (nilpotent) 3-group of index 2.
Manin (1986) used these groups to construct examples of non-abelian CH-quasigroups and to describe the structure of commutative Moufang loops of exponent 3.
Suppose that G is a group that is generated by a conjugacy class D of 3-transpositions and such that the 2 and 3 cores O2(G) and O3(G) are both contained in the center Z(G) of G. Then Fischer (1971) proved that up to isomorphism G/Z(G) is one of the following groups and D is the image of the given conjugacy class: The missing cases with n small above either do not satisfy the condition about 2 and 3 cores or have exceptional isomorphisms to other groups on the list.
The 3-transpositions are of the form x↦x+(x,v)v for non-zero v. The special unitary group SUn(2) has order The projective special unitary group PSUn(2) is the quotient of the special unitary group SUn(2) by the subgroup M of all the scalar linear transformations in SUn(2).
If n=2m is even the two orthogonal groups On±(3) have orders and On+,+(3) ≅ On+,−(3), and On−,+(3) ≅ On−,−(3), because the two classes of transpositions are exchanged by an element of the general orthogonal group that multiplies the quadratic form by a scalar.
The group Onμ,σ(3) has a subgroup of index 2, namely Ωnμ(3), which is simple modulo their centers (which have orders 1 or 2).
If n>4 is odd, and (μ,σ)=(+,+) or (−,−), then Onμ,+(3) and POnμ,+(3) are both isomorphic to SOnμ(3)=Ωnμ(3):2, where SOnμ(3) is the special orthogonal group of the underlying quadratic form Q.
There are numerous degenerate (solvable) cases and isomorphisms between 3-transposition groups of small degree as follows (Aschbacher 1997, p.46): The following groups do not appear in the conclusion of Fisher's theorem as they are solvable (with order a power of 2 times a power of 3).
There are several further isomorphisms involving groups in the conclusion of Fischer's theorem as follows.
Suppose that D is the class of 3-transpositions in G, and d∈D, and let H be the subgroup generated by the set Dd of elements of D commuting with d. Then Dd is a set of 3-transpositions of H, so the 3-transposition groups can be classified by induction on the order by finding all possibilities for G given any 3-transposition group H. For simplicity assume that the derived group of G is perfect (this condition is satisfied by all but the two groups involving triality automorphisms.)
The graph is connected unless the group has a direct product decomposition.
It embodies sufficient relations to define the group Sn.