3-transposition group
In mathematical group theory, a 3-transposition group is a group generated by a conjugacy class of involutions, called the 3-transpositions, such that the product of any two involutions from the conjugacy class has order at most 3. They were first studied by Bernd Fischer (1964, 1970, 1971) who discovered the three Fischer groups as examples of 3-transposition groups.
Fischer (1964) first studied 3-transposition groups in the special case when the product of any two distinct 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 group Sn has order n! and for n>1 has a subgroup An of index 2 that is simple if n>4.
The symmetric group Sn is a 3-transposition group for all n>1. The 3-transpositions are the elements that exchange two points, and leaving each of the remaining points fixed. These elements are the transpositions (in the usual sense) of Sn. (For n=6 there is a second class of 3-transpositions, namely the class of the elements of S6 which are products of 3 disjoint transpositions.)
...
Wikipedia