[GAP Forum] Orthogonal group over finite field

[Derek Holt]

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.