Thank you for the answers I have received ! I will not send mails not related
to GAP any more, sorry for this.
As a bonus I would like to present decomposition of 1451520 elements in O7(Z2)
group which I defined as {A*A^T=I}. Class [a,b,c] means matrix has <a> columns
with 5 ones, <b> columns with 3 ones and <c> columns with one one. The work was
done in GAP, so this would be forgiven in this forum, I hope :)
Counting elements in class [ 0, 0, 7 ]
... it is 5040 <-------- this is permutation subgroup: 7! elements
Counting elements in class [ 0, 4, 3 ]
... it is 176400
Counting elements in class [ 2, 4, 1 ]
... it is 529200
Counting elements in class [ 3, 4, 0 ]
... it is 705600
Counting elements in class [ 6, 0, 1 ]
... it is 35280
Regards,
Marek
"Derek Holt" <d.f.h...@warwick.ac.uk> napisał(a):
> Dear Marek, Dear GAP Forum,
>
> On Tue, Nov 24, 2009 at 09:12:21AM +0100, m...@op.pl wrote:
> > Hello,
> >
> > I have lately interested in finite groups. When I study orthogonal groups
> > over field Z2, I have noticed following issue. Take definition of the
> > orthogonal group:
> > On(F)={A belongs to Mn(F): A*TransposedMat(A)=I}
> > Let's call it natural definition.
>
> This is definitely not the natural definition for fields of characteristic 2!
> In fact, for finite fields of odd characeristic and even dimension there
> are two types of orthogonal group (as you quote from Wilson below), and
> neither is really more natural than the other, so I don't think it a good
> idea to talk about the natural definition over finite fields.
>
> > In dimension 4 and field Z2 there are 48 such matrices. In GAP there are
> > two orthogonal groups in dimension 4: GO(1, 4,2) with 72 elements and
> > GO(-1,4,2) with 120 elements. When I perform following in GAP:
> > g:=GO(1,4,2);
> > gen:=GeneratorsOfGroup(g);
> > Display(gen[1]); Display(gen[2]); Display(gen[1]*TransposedMat(gen[1]));
> > I see that generators do not satisfy condition A*A^T = I.
> >
> > In dimension 5 it seems both definitions GAP and natural gives groups with
> > 720 elements, so the number of elements in the same. Still GAP gives other
> > representation then I expect i.e. generators do not satisfy condition
> > A*A^T = I.
> >
> > Wilson book gives following definition of the orthogonal group (chapter
> > 3.7):
> > "Recall from Section 3.4.6 that, up to equivalence, there are exactly two
> > nonsingular
> > symmetric bilinear forms f on a vector space V over a finite field F
> > of odd order. The orthogonal group O(V, f) is defined as the group of
> > linear
> > maps g satisfying f(ug, vg) = f(u, v) for all u, v from V ."
> >
> > Can somebody explain for me what "orthogonal" means in case of field Z2 ?
> > Why group {A: A*A^T=I} is not "orthogonal" ? Where I can find formula for
> > number of elements in set {A:A*A^T=I} for field Z2 and other fields.
>
> In characteristic 2, the orthogonal groups are only "interesting" in even
> dimension, and they are defined as the groups preserving quadratic forms
> rather than bilinear forms. There are two equivalence classes of such forms,
> so two types of groups. (They are both subgroups of the symplectic group,
> note that symplectic forms are in fact bilinear in characteristic 2.)
> The details are too complicated for an e-mail, and you need to read about
> it in a suitable textbook. In the sentence you quote from Wilson, he is
> referring to odd characteristic - I don't know whether he also deals with
> even characteristic.
>
> But the elements A of GL(n,q) that satisfy A A^T = I do form a subgroup of
> GL(d,q), so it is certainly reasonable to ask what is the structure of
> that subgroup. Let's call it G.
>
> Assume that q is even.
>
> If we let V be the n-dimensional vector space on which GL(n,q) acts, then
> the set of singular vectors under the form defined by the identity matrix
> forms a subspace W of codimension 1 in V, and the orthogonal complement
> X of W has dimension 1. Both of these subspaces are necessarily fixed by G,
> so G is acting reducibly on V. This is why G is not really a natural group to
> study!
>
> If n =2m+1 is odd, then V = W + X, and the form restricted to W is
> symplectic, so G is isomorphic to Sp(2m,q) - you can look of the order of
> that
> in any book dealing with classical groups over finite fields.
>
> For n = 2m even, X < W, and the form induces a symplectic form on the
> 2(m-1) dimensional space W/X. It turns out in this case that G is the
> same group as the stabilizer of a vector in Sp(2m,2), which is a group
> with a normal elementary abelian subgroup of order 2^(2m-1) with
> quotient Sp(2m-2,q). So |G| = 2^(2m-1) |Sp(2m-2,q)|.
> (For m = 2 this gives 8 times 6 = 48.)
>
> For q odd, G is conjugate in GL(n,q) to one of the standard orthogonal groups
> defined by Wilson. There is only one isomorphism class of such groups for
> n odd. For n even, there two types, the +-type and the --type. If I am
> remembering correctly, then G is (conjugate to) the +-type group except
> when d = 2 (mod 4) and q = 3 (mod 4), in which case it is the --type group.
>
> By the way, your question is not really to do with GAP. You might do better
> to ask questions about group theory in a general group theory mailing list,
> such as group pub forum:
> http://people.bath.ac.uk/masgcs/gpf.html
>
> Derek Holt.
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