We have considered two groups : SO(2) and O(2). The second contains the first.

The first contains the neutral element. We can figure the elements of the group as follows :
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.

The elements of the first component form a group (a sub-group).
The elements of the second group do not form a group, for many reasons :

- It does not contain the neutral elements 1.
- we can pick two matrices in this second component, whose product does not belong to this second component. Example :
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The component of the group which contains the neutral element 1 is called the

In the following we will consider groups with 2, 4, 8 components.

The Euclid's group.

We can now integrate this extended, enriched group, to 2d- translation, and we get :
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and the corresponding action of this Euclid's group :
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Suppose we use our group to manipulate, to rule, to study alphabetic letters.

Limit the set to :

 
We have several sizes :
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for their sizes are different. We decide to call their sizes masses so that G and G are similar to particules, objects, atoms, who own different masses. Now, depends on the group which acts on this set of objects. If I use :
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assume this "world" is filled by :
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with a certain spectrum of sizes (masses) and angles. If I operate group actions, whetever they are, I will never find objects which belong to the russian alphabet :
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It will be possible if I take the enriched group, the Euclid's group :
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Then my "world" will become :

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The group has enriched the letters' "zoo". But in my zoo, one is invariant by symmetry, i.e :
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...In general, any symetry with respect to any straight line of the plane, which is a "2d mirror", does not change the "nature" of this character
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I will call this character a "photon" and will assimilate the transform
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to the matter anti-matter duality. Then I get a global zoo :
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We could link letters of same shape (nature) but different sizes (representing their energies), using Descartes'group:
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...But we are not going to build a complete analogical model of elementary particles, based on alphabetical characters. Anyway, you begin to see were we tend to go. Group have very simple aspects, but hidden properties. These properties depend on their sub-groups, which fathers species.

 
...Euclid's group goes with an Euclid's world, with Euclid's zoo. The animals of euclidean geometry are called sphere, cylinder, prisms, plane, straight line, triangles, en so on. The are invariant under some sub-group action. Souriau calls the sub-group linked to a an object, which belongs to a species, the regularity of this object.

 
For an example spheres centered on a given point O are invariant through the sub-group of rotations around this point.

- We can consider that the fact to be invariant is a property of the species called "spheres centered on a point O".

- Conversely we can consider that this property defines the species.