3-transposition group
# 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 $O_2(G)$ and $O_3(G)$ 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 $Fi_{22},$ $Fi_{23},$ $Fi_{24}.$
* G/Z(G) is one of two groups of the form $Ω_8^+(2).S_3$ and $PΩ_8^+(3).S_3,$ where Ω stands for the derived subgroup of the orthogonal group and $S_3$ is the group of diagram automorphisms for the $D_4$ 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 $S_6$ 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 $S_3,$ and for n=2 it is $S_6$ with a simple subgroup of index 2, namely $A_6.$ 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 $S_3$ and for n=3 it has the structure $3^2:Q_8$ $(Q_8$ = 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.

$Fi_{22}$ has order $2^{17}.3^9.5^2.7.11.13$ = 64561751654400 and is simple.

$Fi_{23}$ has order $2^{18}.3^{13}.5^2.7.11.13.17.23$ = 4089470473293004800 and is simple.

$Fi_{24}$ has order $2^{22}.3^{16}.5^2.7^3.11.13.17.23.29$ and has a simple subgroup of index 2, namely $Fi_{24}'.$

## 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 $Q_8$ 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 $3^{1+2}$ denotes the extraspecial group of order 27 and exponent 3, and $Q_8$ 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 $W(D_2)=2^2,$ 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 $O_3(H)$ is not contained in Z(H) then G is the symmetric group $S_5$
* If $O_2(H)$ is not contained in Z(H) then $L=H/O_2(H)$ 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 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 $Fi_{22}$ (and H is an exceptional double cover of $PSU_6(2))$
* If H/Z(H) is of type $Fi_{22}$ then G is of type $Fi_{23}$ and H is a double cover of $Fi_{22}.$
* If H/Z(H) is of type $Fi_{23}$ then G is of type $Fi_{24}$ and H is the product of $Fi_{23}$ and a group of order 2.
* H/Z(H) cannot be of type $Fi_{24}.$

## 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 $S_3$ form a triangle. The 6 transpositions of $S_4$ form an octahedron. The 10 transpositions of $S_5$ 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 $S_n.$

