[GAP Forum] Orthogonal group over finite field

[Frank Lübeck]

On Tue, Nov 24, 2009 at 09:12:21AM +0100, mim_ at op.pl wrote:
> 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. 
> 
> 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. 

Dear Marek Mitros, dear Forum,

Derek Holt has already explained the definition of orthogonal groups and 
given some references.

Let me add that the GAP manual also gives some useful information, ask the
GAP help system for

?GO
?Classical groups

In particular it is mentioned how to find out the corresponding quadratic form
used for a group returned by GO:

gap> g := GO(-1,4,2);
GO(-1,4,2)
gap> form := InvariantQuadraticForm(g);
rec( matrix := <an immutable 4x4 matrix over GF2> )
gap> Display(form.matrix);
 . 1 . .
 . . . .
 . . 1 1
 . . . 1

Furthermore, you find references for the generating sets returned by GO
(and the other classical group constructors).

Best regards,

  Frank Lübeck

-- 
///  Dr. Frank Lübeck, Lehrstuhl D für Mathematik, Templergraben 64,  ///
\\\                    52062 Aachen, Germany                          \\\
///  E-mail: Frank.Luebeck at Math.RWTH-Aachen.De                        ///
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