[GAP Forum] Forum Digest, Vol 191, Issue 8
Horvath Gabor
ghorvath at science.unideb.hu
Sun Nov 3 15:18:54 GMT 2019
Dear Bilal,
If you're interested in groups of order p^4, take a look at this:
https://arxiv.org/pdf/1611.00461.pdf
Without checking this paper thoroughly, it looks to me that your groups
are the 6th and 7th from Table 2, marked by (**). (The third group marked
by (**) has a C5xC5xC5 normal subgroup in it, containing all elements of
order 5, so that cannot be any of your groups.)
Lemma 19 explains why your two groups are in fact nonisomorphic.
Thanks,
Gábor
On Sun, 3 Nov 2019, Bilal Hasanat wrote:
> Dear all, hope that my email finds all of you very well.
> I am a new GAP user. While I am using GAP to find all groups of order 5^4, the obtained list contains 15 groups, say G[i], i=1,2,...,15. I found that G[9] and G[10] have the same StructureDescription (C25 x C5) : C5, although they are not isomorphic groups! . On the other hand, I have try to test why these two groups are not isomorphic using GAP's calculations, and I still find a complete match of what I test for both groups.
> Kindly, is there any way to configure the differences between these two groups using GAP?
> Bilal N. Al-Hasanat Department of MathematicsAl Hussein Bin Talal University
>
> On Thursday, October 24, 2019, 02:00:17 PM GMT+3, forum-request at gap-system.org <forum-request at gap-system.org> wrote:
>
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> Today's Topics:
>
> 1. matrix realization over prime field (Evgeny Vdovin)
> 2. Re: matrix realization over prime field (Frank L?beck)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Thu, 24 Oct 2019 08:20:31 +0700
> From: Evgeny Vdovin <vdovin at math.nsc.ru>
> To: forum at gap-system.org
> Subject: [GAP Forum] matrix realization over prime field
> Message-ID:
> <CAAQ9cL8XLwp3d8wQf8ssVdkvfgBAy6cXeuyMfRGcnUs5YQTRbQ at mail.gmail.com>
> Content-Type: text/plain; charset="UTF-8"
>
> Dear all,
>
> Could you give me an idea, how could I realize the following procedure:
>
> Let A be a n*n matrix over a non-prime field GF(p^k) (say, A in GL(2,4)). I
> need to generate matrix B of size nk*nk over GF(p) such that each k*k block
> in it is an element in GF(p^k) realized as k*k matrices over GF(p) and the
> element corresponds to an element of A.
>
> For example, if
> A =
> [
> [Z(2^2),0*Z(2^2)],
> [0*Z(2^2),Z(2^2)^(-0)]
> ]
> and
> Z(2^2) =
> [
> [a,b],
> [c,d]
> ];
> Z(2^2)^(-1)=
> [
> [x,y],
> [z,t]
> ],
> then
> B=
> [
> [a,b,0*Z(2),0*Z(2)],
> [c,d,0*Z(2),0*Z(2)],
> [0*Z(2),0*Z(2),x,y],
> [0*Z(2),0*Z(2),z,t]
> ].
>
> All the best, Evgeny.
>
> --
> Evgeny Vdovin
> Sobolev Institute of Mathematics
> pr-t Acad. Koptyug, 4
> 630090, Novosibirsk, Russia
> Office +7 383 3297663
> Fax +7 383 3332598
>
>
> ------------------------------
>
> Message: 2
> Date: Thu, 24 Oct 2019 03:55:50 +0200
> From: Frank L?beck <frank.luebeck at math.rwth-aachen.de>
> To: Evgeny Vdovin <vdovin at math.nsc.ru>, forum at gap-system.org
> Subject: Re: [GAP Forum] matrix realization over prime field
> Message-ID: <20191024015550.GI29066 at alkor.math.rwth-aachen.de>
> Content-Type: text/plain; charset=iso-8859-1
>
> Dear Evgeny, dear Forum,
>
> I have written such a function for a demo. It is maybe not very elegant or
> optimized but seems to work:
>
> # write elements of GF(q^d) as dxd-matrices over GF(q)
> MatricesFieldElts := function(q, d)
> local f, bas, basv, z, zmat, res, i;
> f := GF(GF(q), d);
> bas := Basis(f);
> basv := BasisVectors(bas);
> z := Z(q^d);
> zmat := List(basv*z, x-> Coefficients(bas, x));
> for i in zmat do
> ConvertToVectorRep(i, q);
> od;
> MakeImmutable(zmat);
> ConvertToMatrixRep(zmat, q);
> res := [zmat^0];
> for i in [1..q^d-2] do
> res[i+1] := res[i] * zmat;
> od;
> res[q^d] := NullMat(d, d, GF(q));
> return res;
> end;
>
> # blow up GF(q^d)-matrix over subfield of size q and degree d
> BlowUpMatrixOverSmallField := function(mat, q, d)
> local flist, z, f, tmp;
> flist := MatricesFieldElts(q, d);
> z := Z(q^d);
> f := function(c)
> if IsZero(c) then
> return flist[q^d];
> fi;
> return flist[LogFFE(c, z)+1];
> end;
> tmp := List(mat, r-> List(r, f));
> tmp := Concatenation(List(tmp, r-> List([1..d], i-> Concatenation(
> List(r, m-> m[i])))));
> ConvertToMatrixRep(tmp, q);
> return tmp;
> end;
>
> gap> A := [ [ Z(2^2), 0*Z(2) ], [ 0*Z(2), Z(2)^0 ] ];;
> gap> AA := BlowUpMatrixOverSmallField(A, 2, 2);
> <a 4x4 matrix over GF2>
> gap> Display(AA);
> . 1 . .
> 1 1 . .
> . . 1 .
> . . . 1
>
>
> Best regards,
> Frank
>
>
> On Thu, Oct 24, 2019 at 08:20:31AM +0700, Evgeny Vdovin wrote:
>> Dear all,
>>
>> Could you give me an idea, how could I realize the following procedure:
>>
>> Let A be a n*n matrix over a non-prime field GF(p^k) (say, A in GL(2,4)). I
>> need to generate matrix B of size nk*nk over GF(p) such that each k*k block
>> in it is an element in GF(p^k) realized as k*k matrices over GF(p) and the
>> element corresponds to an element of A.
>>
>> For example, if
>> A =
>> [
>> [Z(2^2),0*Z(2^2)],
>> [0*Z(2^2),Z(2^2)^(-0)]
>> ]
>> and
>> Z(2^2) =
>> [
>> [a,b],
>> [c,d]
>> ];
>> Z(2^2)^(-1)=
>> [
>> [x,y],
>> [z,t]
>> ],
>> then
>> B=
>> [
>> [a,b,0*Z(2),0*Z(2)],
>> [c,d,0*Z(2),0*Z(2)],
>> [0*Z(2),0*Z(2),x,y],
>> [0*Z(2),0*Z(2),z,t]
>> ].
>>
>> All the best, Evgeny.
>>
>> --
>> Evgeny Vdovin
>> Sobolev Institute of Mathematics
>> pr-t Acad. Koptyug, 4
>> 630090, Novosibirsk, Russia
>> Office +7 383 3297663
>> Fax +7 383 3332598
>> _______________________________________________
>> Forum mailing list
>> Forum at gap-system.org
>> https://mail.gap-system.org/mailman/listinfo/forum
>
> --
> /// Dr. Frank L?beck, Lehrstuhl D f?r Mathematik, Pontdriesch 14/16,
> \\\ 52062 Aachen, Germany
> /// E-mail: Frank.Luebeck at Math.RWTH-Aachen.De
> \\\ WWW: http://www.math.rwth-aachen.de/~Frank.Luebeck/
>
>
>
> ------------------------------
>
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> End of Forum Digest, Vol 191, Issue 8
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Horvath Gabor
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