[GAP Forum] bug with semidirect product?

[Alexander Konovalov]

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
>>
>>
>>
>>
>>
>>
>>
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>> Forum mailing list
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>> 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