[GAP Forum] 3-transposition groups

[Thomas Breuer]

Dear Marek,

the group GO(+1,6,3) has the structure 2 x PGO(+1,6,3).
The group in question is PGO(+1,6,3), which is called L4(3).2_2
in the ATLAS of Finite Groups.
In GAP, you can construct the group for example as the permutation
group obtained from the projective action of GO(+1,6,3),
or as `PrimitiveGroup( 117, 3 )'.

The group GO(-1,6,3) has the structure 2.U_4(3).(2^2)_{122},
using ATLAS notation.
The group in question is U_4(3).2_2.
In GAP, it can be obtained for example as `PrimitiveGroup( 126, 8 )'.

All the best,
Thomas


On Wed, Apr 10, 2013 at 11:42:05AM +0200, Marek Mitros wrote:
> Dear Forum,
> 
> I have read in wikipedia that there are following 3-transposition groups.
> See below quote from wikipedia article on "3-transposition groups". I have
> no problem to obtain in GAP groups SO(1,n,2), Omega(1,n,2), PSU(n,2),
> Sp(n,2). I can also define Fischer groups Fi22, Fi23, Fi24 using Atlas
> package. I have only trouble with O+-(2n,3) case. E.g. neither SO(1,6,3)
> nor Omega (1,6,3) is 3-transposition group. What is the way to define the
> 3-transposition group defined under case  Oμ, π(*n*, 3) below.
> 
> 
> Regards,
> Marek
> 
> ----------------------------------
> 
> 
> Suppose that *G* is a group that is generated by conjugacy class of
> 3-transpositions and such that the 2 and 3
> cores<http://en.wikipedia.org/wiki/P-core>
> *O*2(*G*) and *O*3(*G*) are both contained in the center *Z*(*G*) of
> *G*and the derived group of
> *G* is perfect. Then Fischer
> (1971<http://en.wikipedia.org/wiki/3-transposition_group#CITEREFFischer1971>)
> 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 *Sn*, and *D* is the class of
>    transpositions.
>    - *G*/*Z*(*G*) is a symplectic group Sp(2*n*, 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 Oμ(2*n*, 2), and *D* is the class
>    of transvections
>    - *G*/*Z*(*G*) is an index 2 subgroup Oμ, π(*n*, 3) of the orthogonal
>    group Oμ(*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<http://en.wikipedia.org/wiki/Fischer_group>Fi
>    22, 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.