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.
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.
Regards,
Marek Mitros
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