[GAP Forum] Forum Digest, Vol 191, Issue 8

[Bilal Hasanat]

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:  
 
 Send Forum mailing list submissions to
    forum at gap-system.org

To subscribe or unsubscribe via the World Wide Web, visit
    https://mail.gap-system.org/mailman/listinfo/forum
or, via email, send a message with subject or body 'help' to
    forum-request at gap-system.org

You can reach the person managing the list at
    forum-owner at gap-system.org

When replying, please edit your Subject line so it is more specific
than "Re: Contents of Forum digest..."


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/



------------------------------

_______________________________________________
Forum mailing list
Forum at gap-system.org
https://mail.gap-system.org/mailman/listinfo/forum


End of Forum Digest, Vol 191, Issue 8
*************************************