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群的组成部分。
我们考虑了两个群：SO(2) 和 O(2)。第二个群包含第一个群。
第一个群包含单位元。我们可以将群的元素表示如下：
(73) 。
第一部分的元素形成一个群（一个子群）。
第二部分的元素不形成一个群，原因有很多：
- 它不包含单位元 1。
- 我们可以在第二部分中选择两个矩阵，它们的乘积不属于第二部分。例如：
(74)
包含单位元 1 的群的部分称为
群的单位元部分。
在接下来的内容中，我们将考虑具有 2、4、8 个部分的群。
欧几里得群。
我们现在可以将这个扩展的、丰富的群整合到二维平移中，得到：
(75)
以及该欧几里得群的相应作用：
(76)
假设我们使用我们的群来操作、控制、研究字母表中的字母。
将集合限制为：A B C D E F G J K L N P Q R S Z
我们有多种尺寸：
(77) A B C D E F G J K L N P Q R S Z A B C D E F G J K L N P Q R S Z
A B C D E F G J K L N P Q R S Z
我们知道无法找到群中的一个元素，或者后续的群作用，能够将：
G 转换为 G
因为它们的尺寸不同。我们决定将它们的尺寸称为“质量”，这样 G 和 G 就类似于具有不同质量的粒子、物体、原子。现在，这取决于作用于这个对象集合上的群。如果我使用：
(78)
假设这个“世界”被：
(79)
某种尺寸（质量）和角度的谱所填充。如果我应用任何群作用，我都永远不会找到属于俄语字母表的对象：
(80)
如果我使用扩展的群，即欧几里得群，这将变得可能：
(80b)
那么我的“世界”将变成：
(81)
这个群丰富了字母的“动物园”。但在我的动物园中，有一个元素是通过对称性保持不变的，即：
(82)
(83)
(84)
(85)
...一般来说，任何关于平面中直线的对称性，即“二维镜面”，都不会改变该字符的“性质”
(86)
我将把这个字符称为“光子”，并把变换
(87)
等同于物质与反物质的二元性。那么我将得到一个整体的动物园：
(88)
我们可以使用笛卡尔群将相同形状（性质）但不同尺寸（代表它们的能量）的字母联系起来：
(89)
...但我们不会构建一个基于字母表字符的完整粒子模型。无论如何，你已经开始看到我们想要达到的方向。群有非常简单的方面，但也有隐藏的性质。这些性质取决于它们的子群，这些子群生成了种类。
...欧几里得群伴随着一个欧几里得世界，一个欧几里得动物园。欧几里得几何的动物被称为球体、圆柱体、棱柱、平面、直线、三角形等。它们在某些子群的作用下是不变的。Souriau 将与属于某一类别的对象相关的子群称为该对象的 规则。
例如，以给定点 O 为中心的球体在围绕该点的旋转子群作用下保持不变。
- 我们可以认为不变性是称为“以点 O 为中心的球体”的种类的一个属性。
- 反过来，我们可以认为这个属性 定义 了该种类。
原始版本（英文）
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Components of a group.
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 :
(73) .
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 :
(74)
The component of the group which contains the neutral element 1 is called the
neutral component of the group.
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 :
(75)
and the corresponding action of this Euclid's group :
(76)
Suppose we use our group to manipulate, to rule, to study alphabetic letters.
Limit the set to : A B C D E F G J K L N P Q R S Z
We have several sizes :
(77) A B C D E F G J K L N P Q R S Z A B C D E F G J K L N P Q R S Z
A B C D E F G J K L N P Q R S Z
We know that we cannot find any element of the group, and a subsequent group's action, which can transform :
G into G
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 :
(78)
assume this "world" is filled by :
(79)
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 :
(80)
It will be possible if I take the enriched group, the Euclid's group :
(80b)
Then my "world" will become :
(81)
The group has enriched the letters' "zoo". But in my zoo, one is invariant by symmetry, i.e :
(82)
(83)
(84)
(85)
...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
(86)
I will call this character a "photon" and will assimilate the transform
(87)
to the matter anti-matter duality. Then I get a global zoo :
(88)
We could link letters of same shape (nature) but different sizes (representing their energies), using Descartes'group:
(89)
...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.