[GAP Forum] bug with semidirect product?

[Rudolf Zlabinger]

Dear Alexander,

after a time ago I wrote this message, I saw that the description of the 
reference manual is a little bit misleading and my considerations was a 
little bit "hair splitting" on the other hand. If read in this way:

"If N is a full row space over a field F, alpha must be a homomorphism from 
G into a matrix group of the right dimension over a subfield of F, or into a 
permutation group (in this case permutation matrices are taken).
In the second variant, autgp must be a group of automorphism of N, it is a 
shorthand for SemidirectProduct(autgp,IdentityMapping(autgp),N). " (except 
for the case above, where autgp is to be interpreted as a matrix group)

there is no need for autgp to be an automorphism group, as N is a full row 
space in our case and the last sentence:

"Note that (unless autgrp has been obtained by the operation 
AutomorphismGroup) you have to test IsGroupOfAutomorphisms(autgrp) to ensure 
that GAP knows that autgrp consists of group automorphisms."

doesnt apply.

Thank you for your explanation nevertheless, all the best, Rudolf



----- Original Message ----- 

From: "Alexander Konovalov" <alexander.konovalov at gmail.com>
To: "Rudolf Zlabinger" <Rudolf.Zlabinger at chello.at>
Cc: "GAP Forum" <forum at gap-system.org>
Sent: Wednesday, July 18, 2007 12:44 PM
Subject: Re: [GAP Forum] bug with semidirect product?


> Dear Rudolf,
>
> I think that here you misinterpreted the meaning of 
> IsGroupOfAutomorphisms.
> This is an example of its intended usage:
>
> gap> G:=DihedralGroup(8);
> <pc group of size 8 with 3 generators>
> gap> A:=AutomorphismGroup(G);
> <group of size 8 with 3 generators>
> gap> x:=GeneratorsOfGroup(A)[1];
> Pcgs([ f1, f2, f3 ]) -> [ f1*f2, f2, f3 ]
> gap> H:=Group(x);
> <group with 1 generators>
> gap> IsGroupOfAutomorphisms(H);
> true
>
> Thus, H s a group whose elements are group automorphisms - without  any 
> relation
> to the fact whether H is a (full) automorphism group of some other  group 
> or not.
>
> Best wishes,
> Alexander
>
>
> On 19 Jun 2007, at 23:17, Rudolf Zlabinger wrote:
>
>> Dear Forum,
>>
>> there is something wrong with that semidirect product in general:
>>
>> The Reference Manual says for SemidirectProduct(autgrp,N) (the  second 
>> variant), as used in our case:
>>
>> In the second variant, autgp must be a group of automorphism of N,  it is 
>> a shorthand for SemidirectProduct(autgp,IdentityMapping (autgp),N). Note 
>> that (unless autgrp has been obtained by the  operation 
>> AutomorphismGroup) you have to test IsGroupOfAutomorphisms (autgrp) to 
>> ensure that GAP knows that autgrp consists of group  automorphisms.
>>
>> As the conditions dont hold in the following way:
>>
>> gap> v:=GF(9)^2;
>> ( GF(3^2)^2 )
>> gap> g:=GL(2,9);
>> GL(2,9)
>> gap> IsGroupOfAutomorphisms(g);
>> false
>> gap>
>>
>> the result is not well defined by function description, and may be  wrong 
>> in any way.
>>
>> best regards, Rudolf Zlabinger
>>
>>
>>
>>
>>
>>
>>
>> ----- Original Message ----- From: "Stefan Kohl" 
>> <kohl at mathematik.uni-stuttgart.de>
>> To: "GAP Forum" <forum at gap-system.org>
>> Sent: Tuesday, June 19, 2007 1:38 PM
>> Subject: Re: [GAP Forum] bug with semidirect product?
>>
>>
>>> Dear Forum,
>>>
>>> Vdovin Evgeni wrote:
>>>
>>>> When I try to construct semidirect product GL_2(9)*GF(9)^2, it  returns 
>>>> a group with GF(3)^2 as a normal subgroup (see the  listing below).
>>>>
>>>> gap> V:=GF(9)^2;
>>>> ( GF(3^2)^2 )
>>>> gap> G:=GeneralLinearGroup(2,9);
>>>> GL(2,9)
>>>> gap> p:=SemidirectProduct(G,V);
>>>> <matrix group of size 466560 with 3 generators>
>>>> gap> L:=Image(Embedding(p,1));
>>>> Group(
>>>> [ [ [ Z(3^2), 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z (3), 
>>>> 0*Z(3),
>>>>           Z(3)^0 ] ],
>>>>  [ [ Z(3), Z(3)^0, 0*Z(3) ], [ Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 
>>>> 0*Z(3),
>>>>           Z(3)^0 ] ] ])
>>>> gap> U:=Image(Embedding(p,2));
>>>> Group(
>>>> [ [ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3) ], [ Z(3) ^0, 
>>>> 0*Z(3),
>>>>           Z(3)^0 ] ],
>>>>  [ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3) ],
>>>>      [ 0*Z(3), Z(3)^0, Z(3)^0 ] ] ])
>>>> gap> Order(U);
>>>> 9
>>>>
>>>> Can anybody explain, what is wrong here?
>>>
>>> Thanks for the report.
>>>
>>> In fact this is a known bug, which will be fixed in the next release:
>>>
>>> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
>>> %%%%%%%%%%%
>>> ! Date
>>> 2007/01/17
>>> ! Changed by
>>> AH
>>> ! Reported by
>>> anvita21
>>> ! Type of Change
>>> Fix: wrong result
>>> ! Description
>>> When forming the semidirect product of a matrix group with a  vector 
>>> space
>>> over a non-prime field
>>> the embedding of the vector space gives a wrong result.
>>> ! Changed Files
>>> lib/gprd.gi
>>> ! End
>>> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
>>> %%%%%%%%%%%
>>>
>>> If you need a workaround for this bug already now, then please  write to
>>> support at gap-system.org .
>>>
>>> Best wishes,
>>>
>>>     Stefan Kohl
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> _______________________________________________
>>> Forum mailing list
>>> Forum at mail.gap-system.org
>>> http://mail.gap-system.org/mailman/listinfo/forum
>>
>>
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>
> --
> Dr. Alexander Konovalov               School of Computer Science
> & Centre for Interdisciplinary Research in Computational Algebra
>              University of St Andrews    Tel +44/0 (1334) 461633
> http://www.cs.st-andrews.ac.uk/~alexk    Fax +44/0 (1334) 463278
>
>
>