# [GAP Forum] Re: Forum Digest, Vol 12, Issue 4

taoufik karkar taoufik.karkar at fst.rnu.tn
Wed Dec 1 15:13:09 GMT 2004

```**** Forum archive pruned to remove irrelevant parts of digest message.
**** John McDermott, GAP Forum mailing list administrator, 9-Dec-2004.

> Objet: Re: [GAP Forum] parabolic and Borel subgroups
> Date: Wed, 24 Nov 2004 06:28:03 -0500
> De: David Joyner <wdj at usna.edu>
> A: forum at gap-system.org
> Copies à: David Joyner <wdj at usna.edu>
> Références: <auto-000018098111 at maktoobchat.net>
>
>
> p:=7;
> G:=SL(2,p);
> a0:=[ [ Z(p), Z(p) ], [ Z(p)*0, Z(p)^(-1) ] ];
> n0:=[ [ Z(p)0, Z(p) ], [ Z(p)*0, Z(p)0 ] ];
> B:=Group([a0,n0]);
> IsSubgroup(G,B);
>
> On the other hand, it is convenient often to first convert the group
> SL(2,7) into a permutation group. For this use,
>
> G:=SL(2,p);
> iso:=IsomorphismPermGroup(G);
> G0:=Image(iso);
> a0:=Image(iso,[ [ Z(p), Z(p) ], [ Z(p)*0, Z(p)^(-1) ] ]);
> n0:=Image(iso,[ [ Z(p)0, Z(p) ], [ Z(p)*0, Z(p)0 ] ]);
> B:=Group([a0,n0]);; # the Borel as a permutation group
>
> Hope this helps. - David Joyner
>
> ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
>
>
> >Dear forum,
> >
> >Is there any way to get the parabolic subgroups and the Borel subgroups of
> >SL(2,7)? How?
> >
> >
> >
> >Thanks
> >
> >
> >
> >Univ. Of Science And Technology Of China,Dept Of Math.
> >
> >Tel No : +86  551_3650275 -------------------
> >
> >
> >_________________________________________________
For  groups like  GL(2,K),SL(2,K), PSL(2,K) all parabolic  subgroups are
Borel subgroups.
Borel subgroups of these groups are the  conjugates of one triangular
matrices subgroup.
For groups like GL(n,K), where n is greater than 2, there are a lot of
types of parabolic subgroups.

taoufik karkar
dept math
faculty of sciences of tunis
taoufik.karkar at fst.rnu.tn

```