3-transposition group

In mathematical group theory, a 3-transposition group is a group generated by a class of involutions such that the product of any two has order at most 3. They were first studied by Bernd Fischer (1964, 1971) who discovered the three Fischer groups as examples of 3-transposition groups.

History

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.

Fischer's theorem

Suppose that G is a group that is generated by conjugacy class 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 and the derived group of G is perfect. 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:

• G/Z(G) is a symmetric group S_n, and D is the class of transpositions.
• G/Z(G) is a symplectic group Sp(2n, 2) over the field of order 2, and D is the class of transvections
• G/Z(G) is a projective special unitary group PSU(n, 2), and D is the class of transvections
• G/Z(G) is an orthogonal group O^μ(2n, 2), and D is the class of transvections
• G/Z(G) is an index 2 subgroup O^{μ, π}(n, 3) of the orthogonal group O^μ(n, 3) generated by the class D of reflections of norm π vectors, where μ and π can be 1 or –1.
• G/Z(G) is one of the three Fischer groups Fi₂₂, Fi₂₃, Fi₂₄.

If the condition that the derived group of G is perfect is dropped there are two extra cases:

• G/Z(G) is one of two groups containing on orthogonal group O⁺(8, 2) or O^–(8, 3) with index 3.

The idea of the proof is 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 D_d of elements of D commuting with d. Then D_d 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.

• If O₃(H) is not contained in Z(H) then G is the symmetric group S₅
• If O₂(H) is not contained in Z(H) then L=H/O₂(H) is a 3-transposition group, and L/z(L) is either of type Sp(2n, 2) in which case G/Z(G) is of type Sp(2n+2, 2), or of type PSU(n, 2) in which case G/Z(G) is of type PSU(n+2, 2)
• If H/Z(H) is of type S_n then either G is of type S_n+2 or n = 6 and G is of type O^–(6, 2)
• If H/Z(H) is of type Sp(2n, 2) with n ≥ 6 then G is of type O^μ(2n+2, 2)
• H/Z(H) cannot be of type O^μ(2n, 2) for n ≥ 4
• If H/Z(H) is of type O^{μ, π}(n, 3) with n ≥ 5 then G is of type O^{–μπ, π}(n+1, 3)
• If H/Z(H) is of type PSU(n, 2) for n ≥ 5 then n = 6 and G is of type Fi₂₂
• If H/Z(H) is of type Fi₂₂ then G is of type Fi₂₃
• If H/Z(H) is of type Fi₂₃ then G is of type Fi₂₄
• H/Z(H) cannot be of type Fi₂₄.

3-transpositions and graph theory

It is fruitful to treat 3-transpositions as vertices of a graph. Join the pairs that do not commute, i. e. have a product of order 3. The graph is connected unless the group has a direct product decomposition. The graphs corresponding to the smallest symmetric groups are familiar graphs. The 3 transpositions of S₃ form a triangle. The 6 transpositions of S₄ form an octahedron. The 10 transpositions of S₅ form the complement of the Petersen graph.

The symmetric group S_n can be generated by n–1 transpositions: (1 2), (2 3), ..., (n−1 n) and the graph of this generating set is a straight line. It embodies sufficient relations to define the group S_n.^[1]

References

1. ↑ L. E. Dickson, 'Linear Groups' (1900), p. 287.

• Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics 124, Cambridge University Press, ISBN 978-0-521-57196-8, MR 1423599  contains a complete proof of Fischer's theorem.
• Fischer, Bernd (1964), "Distributive Quasigruppen endlicher Ordnung", Mathematische Zeitschrift 83: 267–303, doi:10.1007/BF01111162, ISSN 0025-5874, MR 0160845 
• Fischer, Bernd (1971), "Finite groups generated by 3-transpositions. I", Inventiones Mathematicae 13: 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).
• Manin, Yuri Ivanovich (1986) [1972], Cubic forms, North-Holland Mathematical Library 4 (2nd ed.), Amsterdam: North-Holland, ISBN 978-0-444-87823-6, MR 833513