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,
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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 <vdo...@math.nsc.ru>
To: forum@gap-system.org
Subject: [GAP Forum] matrix realization over prime field
Message-ID:
<caaq9cl8xlwp3d8wqf8ssvdkvfgbay6cxeuymfrgcnus5yqt...@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.lueb...@math.rwth-aachen.de>
To: Evgeny Vdovin <vdo...@math.nsc.ru>, forum@gap-system.org
Subject: Re: [GAP Forum] matrix realization over prime field
Message-ID: <20191024015550.gi29...@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@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.lueb...@math.rwth-aachen.de
\\\ WWW: http://www.math.rwth-aachen.de/~Frank.Luebeck/
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