# [GAP Forum] parabolic and Borel subgroups

John McDermott jmcd at dcs.st-and.ac.uk
Thu Dec 9 11:15:41 GMT 2004

```Dear Forum,

Last week, a message with the subject line

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

was sent to the Forum. This message included by accident an entire
Forum mailing list digest, so was a little too long to be easy to read.
To clarify, the relevant part of that message is below.

John.

Begin forwarded message:

>>
>> 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

--
John McDermott
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```