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.

History

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.

Fischer's theorem

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:

• G/Z(G) is the trivial group.
• G/Z(G) is a symmetric group S_n for n≥5, and D is the class of transpositions. (If n=6 there is a second class of 3-transpositions).
• G/Z(G) is a symplectic group Sp_2n(2) with n≥3 over the field of order 2, and D is the class of transvections. (When n=2 there is a second class of transpositions.)
• G/Z(G) is a projective special unitary group PSU_n(2) with n≥5, and D is the class of transvections
• G/Z(G) is an orthogonal group O^μ_2n(2) with μ=±1 and n≥4, and D is the class of transvections
• G/Z(G) is an index 2 subgroup PO_n^μ,+(3) of the projective orthogonal group PO_n^μ(3) (with μ=±1 and n≥5) generated by the class D of reflections of norm +1 vectors.
• G/Z(G) is one of the three Fischer groups Fi₂₂, Fi₂₃, Fi₂₄.
• G/Z(G) is one of two groups of the form Ω₈⁺(2).S₃ and PΩ₈⁺(3).S₃, where Ω stands for the derived subgroup of the orthogonal group and S₃ is the group of diagram automorphisms for the D₄ Dynkin diagram.

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.

Important examples

The group S_n has order n! and for n>1 has a subgroup A_n of index 2 that is simple if n>4.

The symmetric group S_n 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 S_n. (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.)

The symplectic group Sp_2n(2) has order

$2^{n^{2}}(2^{2}-1)(2^{4}-1)\cdots (2^{2n}-1)$

It is a 3-transposition group for all n≥1. It is simple if n>2, while for n=1 it is S₃, and for n=2 it is S₆ with a simple subgroup of index 2, namely A₆. The 3-transpositions are of the form x↦x+(x,v)v for non-zero v.

The special unitary group SU_n(2) has order

$2^{n(n-1)/2}(2^{2}-1)(2^{3}+1)\cdots (2^{n}-(-1)^{n})$

The projective special unitary group PSU_n(2) is the quotient of the special unitary group SU_n(2) by the subgroup M of all the scalar linear transformations in SU_n(2). The subgroup M is the center of SU_n(2). Also, M has order gcd(3,n).

The group PSU_n(2) is simple if n>3, while for n=2 it is S₃ and for n=3 it has the structure 3²:Q₈ (Q₈ = quaternion group).

Both SU_n(2) and PSU_n(2) are 3-transposition groups for n=2 and for all n≥4. The 3-transpositions of SU_n(2) for n=2 or n≥4 are of the form x↦x+(x,v)v for non-zero vectors v of zero norm. The 3-transpositions of PSU_n(2) for n=2 or n≥4 are the images of the 3-transpositions of SU_n(2) under the natural quotient map from SU_n(2) to PSU_n(2)=SU_n(2)/M.

The orthogonal group O_2n^±(2) has order

$2\times 2^{n(n-1)}(2^{2}-1)(2^{4}-1)\cdots (2^{2n-2}-1)(2^{n}\mp 1)$

(Over fields of characteristic 2, orthogonal group in odd dimensions are isomorphic to symplectic groups.) It has an index 2 subgroup (sometimes denoted by Ω_2n^±(2)), which is simple if n>2.

The group O_2n^μ(2) is a 3-transposition group for all n>2 and μ=±1. The 3-transpositions are of the form x↦x+(x,v)v for vectors v such that Q(v)=1, where Q is the underlying quadratic form for the orthogonal group.

The orthogonal groups O_n^±(3) are the automorphism groups of quadratic forms Q over the field of 3 elements such that the discriminant of the bilinear form (a,b)=Q(a+b)−Q(a)−Q(b) is ±1. The group O_n^μ,σ(3), where μ and σ are signs, is the subgroup of O_n^μ(3) generated by reflections with respect to vectors v with Q(v)=+1 if σ is +, and is the subgroup of O_n^μ(3) generated by reflections with respect to vectors v with Q(v)=-1 if σ is −.

For μ=±1 and σ=±1, let PO_n^μ,σ(3)=O_n^μ,σ(3)/Z, where Z is the group of all scalar linear transformations in O_n^μ,σ(3). If n>3, then Z is the center of O_n^μ,σ(3).

For μ=±1, let Ω_n^μ(3) be the derived subgroup of O_n^μ(3). Let PΩ_n^μ(3)= Ω_n^μ(3)/X, where X is the group of all scalar linear transformations in Ω_n^μ(3). If n>2, then X is the center of Ω_n^μ(3).

If n=2m+1 is odd the two orthogonal groups O_n^±(3) are isomorphic and have order

$2\times 3^{m^{2}}(3^{2}-1)(3^{4}-1)\cdots (3^{2m}-1)$

and O_n^+,+(3) ≅ O_n^−,−(3) (center order 1 for n>3), and O_n^−,+(3) ≅ O_n^+,−(3) (center order 2 for n>3), because the two quadratic forms are scalar multiples of each other, up to linear equivalence.

If n=2m is even the two orthogonal groups O_n^±(3) have orders

$2\times 3^{m(m-1)}(3^{2}-1)(3^{4}-1)\cdots (3^{2m-2}-1)(3^{m}\mp 1)$

and O_n^+,+(3) ≅ O_n^+,−(3), and O_n^−,+(3) ≅ O_n^−,−(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. If n=2m, m>1 and m is even, then the centre of O_n^+,+(3) ≅ O_n^+,−(3) has order 2, and the centre of O_n^−,+(3) ≅ O_n^−,−(3) has order 1. If n=2m, m>2 and m is odd, then the centre of O_n^+,+(3) ≅ O_n^+,−(3) has order 1, and the centre of O_n^−,+(3) ≅ O_n^−,−(3) has order 2.

If n>3, and μ=±1 and σ=±1, the group O_n^μ,σ(3) is a 3-transposition group. The 3-transpositions of the group O_n^μ,σ(3) are of the form x↦x−(x,v)v/Q(v)=x+(x, v)/(v,v) for vectors v with Q(v)=σ, where Q is the underlying quadratic form of O_n^μ(3).

If n>4, and μ=±1 and σ=±1, then O_n^μ,σ(3) has index 2 in the orthogonal group O_n^μ(3). The group O_n^μ,σ(3) has a subgroup of index 2, namely Ω_n^μ(3), which is simple modulo their centers (which have orders 1 or 2). In other words, PΩ_n^μ(3) is simple.

If n>4 is odd, and (μ,σ)=(+,+) or (−,−), then O_n^μ,+(3) and PO_n^μ,+(3) are both isomorphic to SO_n^μ(3)=Ω_n^μ(3):2, where SO_n^μ(3) is the special orthogonal group of the underlying quadratic form Q. Also, Ω_n^μ(3) is isomorphic to PΩ_n^μ(3), and is also non-abelian and simple.

If n>4 is odd, and (μ,σ)=(+,−) or (−,+), then O_n^μ,+(3) is isomorphic to Ω_n^μ(3)×2, and O_n^μ,+(3) is isomorphic to Ω_n^μ(3). Also, Ω_n^μ(3) is isomorphic to PΩ_n^μ(3), and is also non-abelian and simple.

If n>5 is even, and μ=±1 and σ=±1, then O_n^μ,+(3) has the form Ω_n^μ(3):2, and PO_n^μ,+(3) has the form PΩ_n^μ(3):2. Also, PΩ_n^μ(3) is non-abelian and simple.

Fi₂₂ has order 2¹⁷.3⁹.5².7.11.13 = 64561751654400 and is simple.

Fi₂₃ has order 2¹⁸.3¹³.5².7.11.13.17.23 = 4089470473293004800 and is simple.

Fi₂₄ has order 2²².3¹⁶.5².7³.11.13.17.23.29 and has a simple subgroup of index 2, namely Fi₂₄'.

Isomorphisms and solvable cases

There are numerous degenerate (solvable) cases and isomorphisms between 3-transposition groups of small degree as follows (Aschbacher 1997, p.46):

Solvable groups

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).

$S_{1}=SU_{1}(2)=PSU_{1}(2)=O_{1}^{\pm ,\pm }(3)=PO_{1}^{\pm ,\pm }(3)=PO_{1}^{\pm ,\mp }(3)$ has order 1.

$S_{2}=O_{2}^{+}(2)=O_{1}^{\pm ,\mp }(3)=O_{2}^{+,\pm }(3)=PO_{2}^{+,\pm }(3)=PO_{2}^{-,\pm }(3)$ has order 2, and it is a 3-transposition group.

$O_{2}^{-,\pm }(3)=PO_{3}^{\pm ,\mp }(3)=2^{2}$ is elementary abelian of order 4, and it is not a 3-transposition group.

$S_{3}=Sp_{2}(2)=SU_{2}(2)=PSU_{2}(2)=O_{2}^{-}(2)$ has order 6, and it is a 3-transposition group.

$O_{3}^{\pm ,\mp }(3)=2^{3}$ is elementary abelian of order 8, and it is not a 3-transposition group.

$S_{4}=O_{3}^{\pm ,\pm }(3)=PO_{3}^{\pm ,\pm }(3)$ has order 24, and it is a 3-transposition group.

$PSU_{3}(2)=3^{2}:Q_{8}$ has order 72, and it is not a 3-transposition group, where Q₈ denotes the quaternion group.

$O_{4}^{+}(2)=(S_{3}\times S_{3}):2$ has order 72, and it is not a 3-transposition group.

$SU_{3}(2)=3^{1+2}:Q_{8}$ has order 216, and it is not a 3-transposition group, where 3¹⁺² denotes the extraspecial group of order 27 and exponent 3, and Q₈ denotes the quaternion group.

$PO_{4}^{+,\pm }(3)=(A_{4}\times A_{4}):2$ has order 288, and it is not a 3-transposition group.

$O_{4}^{+,\pm }(3)=(SL_{2}(3)*SL_{2}(3)):2$ has order 576, where * denotes the non-direct central product, and it is not a 3-transposition group.

Isomorphisms

There are several further isomorphisms involving groups in the conclusion of Fischer's theorem as follows. This list also identifies the Weyl groups of ADE Dynkin diagrams, which are all 3-transposition groups except W(D₂)=2², with groups on Fischer's list (W stands for Weyl group).

$S_{5}=O_{4}^{-}(2)$ has order 120, and the group is a 3-transposition group.

$S_{6}=Sp_{4}(2)=O_{4}^{-,\pm }(3)=PO_{4}^{-,\pm }(3)$ has order 720 (and 2 classes of 3-transpositions), and the group is a 3-transposition group.

$S_{8}=O_{6}^{+}(2)=$ has order 40320, and the group is a 3-transposition group.

$W(E_{6})=O_{6}^{-}(2)=O_{5}^{\pm ,\pm }(3)=PO_{5}^{\pm ,\pm }(3)$ has order 51840, and the group is a 3-transposition group.

$PO_{5}^{\pm ,\mp }(3)=SU_{4}(2)=PSU_{4}(2)$ has order 25920, and the group is a 3-transposition group.

$W(E_{7})=2\times Sp_{6}(2)$ has order 2903040, and the group is a 3-transposition group.

$W(E_{8})=2.O_{8}^{+}(2)$ has order 69672960, and the group is a 3-transposition group.

$O_{4s}^{+,+}(3)=O_{4s}^{+,-}(3)=2.PO_{4s}^{+,+}(3)=2.PO_{4s}^{+,-}(3)$ for all s≥1, and the group is a 3-transposition group if s≥2.

$O_{4s}^{-,+}(3)=O_{4s}^{-,-}(3)=PO_{4s}^{-,+}(3)=PO_{4s}^{-,-}(3)$ for all s≥1, and the group is a 3-transposition group for all s≥1.

$O_{4s+2}^{+,+}(3)=O_{4s+2}^{+,-}(3)=2.PO_{4s+2}^{+,+}(3)=2.PO_{4s+2}^{+,-}(3)$ for all s≥0, and the group is a 3-transposition group for all s≥0.

$O_{4s+2}^{-,+}(3)=O_{4s+2}^{-,-}(3)=PO_{4s+2}^{-,+}(3)=PO_{4s+2}^{-,-}(3)$ for all s≥0, and the group is a 3-transposition group if s≥1.

$O_{2m+1}^{+,+}(3)=O_{2m+1}^{-,-}(3)=PO_{2m+1}^{+,+}(3)=PO_{2m+1}^{-,-}(3)$ for all m≥0, and the group is a 3-transposition group if m≥1.

$O_{2m+1}^{-,+}(3)=O_{2m+1}^{+,-}(3)=2\times PO_{2m+1}^{-,+}(3)=2\times PO_{2m+1}^{+,-}(3)$ for all m≥0, and the group is a 3-transposition group if m=0 or m≥2.

$W(A_{n})=S_{n+1}$ for all n≥1, and the group is a 3-transposition group for all n≥1.

$W(D_{n})=2^{n-1}.S_{n}$ for all n≥2, and the group is a 3-transposition group if n≥3.

Proof

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. For simplicity assume that the derived group of G is perfect (this condition is satisfied by all but the two groups involving triality automorphisms.)

• 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 O₆⁻(2)
• If H/Z(H) is of type Sp_2n(2) with 2n ≥ 6 then G is of type O_2n+2^μ(2)
• H/Z(H) cannot be of type O_2n^μ(2) for n ≥ 4.
• If H/Z(H) is of type PO_n^{μ, π}(3) for n>4 then G is of type PO_n+1^{−μπ, π}(3).
• If H/Z(H) is of type PSU_n(2) for n ≥ 5 then n = 6 and G is of type Fi₂₂ (and H is an exceptional double cover of PSU₆(2))
• If H/Z(H) is of type Fi₂₂ then G is of type Fi₂₃ and H is a double cover of Fi₂₂.
• If H/Z(H) is of type Fi₂₃ then G is of type Fi₂₄ and H is the product of Fi₂₃ and a group of order 2.
• 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.

References

• Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics, vol. 124, Cambridge University Press, ISBN 978-0-521-57196-8, MR 1423599, archived from the original on 2016-03-04, retrieved 2010-12-06 contains a complete proof of Fischer's theorem.
• Fischer, Bernd (1964), "Distributive Quasigruppen endlicher Ordnung", Mathematische Zeitschrift, 83 (4): 267–303, doi:10.1007/BF01111162, ISSN 0025-5874, MR 0160845, S2CID 123008891
• Fischer, Bernd (1970), Finite groups generated by 3-transpositions, preprint, Coventry: Mathematics Institute, University of Warwick The first part of this preprint (4 of 19 sections) was published as Fischer, Bernd (1971), "Finite groups generated by 3-transpositions. I", Inventiones Mathematicae, 13 (3): 232–246, Bibcode:1971InMat..13..232F, doi:10.1007/BF01404633, MR 0294487, S2CID 120817150 The later part with the construction of the Fischer groups is still unpublished (as of 2014).
• Manin, Yuri Ivanovich (1986) [1972], Cubic forms, North-Holland Mathematical Library, vol. 4 (2nd ed.), Amsterdam: North-Holland, ISBN 978-0-444-87823-6, MR 0833513
• Weiss, Richard (1983), "On Fischer's characterization of Sp_2n(2) and U_n(2)", Communications in Algebra, 11 (22): 2527–54, doi:10.1080/00927878308822979, MR 0733341
• Weiss, Richard (1985), "A uniqueness lemma for groups generated by 3-transpositions", Mathematical Proceedings of the Cambridge Philosophical Society, 97 (3): 421–431, Bibcode:1985MPCPS..97..421W, doi:10.1017/S030500410006299X, MR 0778676, S2CID 123397959