Take a square matrix a. The transposed matrix corresponds to the commutation of the terms which are symmetric with respect to the diagonal, as shown on the figure :
(38)

We write the inverse matrix a^-1
It obeys :

Right now, we will no longer write the sign x and just write :

 (38b)

One can show that :
(38c)

 
so that the determinant of an orthogonal matrix is ± 1 .
They are orthogonal matrixes of any rank ( n,n) . They form groups  

 
Consider matrixes :
(39)

They are orthogonal matrixes, whose determinant is :

It is a sub-group of the orthogonal group O(2), which is called "special orthogonal group" SO(2).
We have an orthogonal group O(3), composed by (3,3) orthogonal matrixes, whose determinant = ± 1 . It owns a sub-group SO(3) composed by orthogonal matrixes whose determinant is + 1 .

In four dimensions : we have the orthogonal group O(4) and its sub-group : the special orthogonal group SO(4).

n dimensions : orthogonal group O(n), composed by (n,n) orthogonal matrixes, whose determinant is ± 1 . It owns a sub group, called Special orthogonal SO(n) limited to orthogonal matrixes whose determinant is + 1.

One can show that the dimension of an orthogonal group is
(40)

Applying to two dimensional space : the dimension of the group is 1.
Applying to three dimensional space, the dimension of the group is three ( the three Euler's angles ).
Applying to four dimensional space, the dimension becomes six.
We have introduced the oriented Special Eulclid's group SE(2):
(41)

Which combined rotations and translation.
Call :
(42)

 
Then we can write the matrix and the action on space :
(43)

Remark :
(44)

In our 2d flat space, in our plane, we find objects like :
(45)

Considering these peculiar objects :
(46)

they belong to a species. If I take any couple of those object, I can find and element of the group which carries the first onto the second, and vice-versa.
The second sub-set of object :
(47)

belongs to another species.
The third, too :
(48)

But :
(49)

I cannot fing any combination rotation a plus translation c that put one on the other.
Can we modify oriented Euclid's group in order to make this possible ?