[GAP Forum] Orthogonal group over finite field

[mim_ at op.pl]

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.Holt at warwick.ac.uk> napisał(a): 
 > Dear Marek, Dear GAP Forum,
 > 
 > On Tue, Nov 24, 2009 at 09:12:21AM +0100, mim_ at 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.