Vladimir I. Arnold ODE'S book, about action group

[Martin T]

hi everyone, I'm electrical engineer student and i like a lot arnold's book of ordinary differential equations (3rd), but i have a gap about how defines action group for a group and from an element of the group.For example Artin's algebra book get another definition also Vinberg's algebra book has another focus. Why different definitions are equivallent?. tips about arnold's book?. THANKS!
Arnold say:
A transformation of a set is a one-to-one mapping of the set onto itself(a bijective).
A collection of transformations of a set is called a transformation group if it contains the inverse of each of its transformations and the product of any two of its transformations.
Let A be a transformation group on the set X. Multiplication and inversion define mappings A × A→A and A→A, ( the pair (f,g) goes to fg, and the element g to g^-1. A set endowed with these two mappings is called an abstract group.Thus a group is obtaing from a transformation group ignoring the set (X) that is transformed.

Let M be a group and M a set. We say that an action of the group G on the set M is defined if to each element g of G there corresponds a transformation Tg : M→M of the set M, to the product and inverse elements corresponds Tfg=Tf Tg, Tg^-1=(Tg)^-1.
Each transformation group of a set naturally acts on that set (Tg ≡ g), but may also act on other sets.
The transformation Tg is also called the action of the element g of the group G on M. The action of the group G on M defines another mapping T: G × M → M assingning to the pair g,m the point Tgm.
If the action is fixed, then the result Tgm of the action of the element g on a point m is denoted gm for short.Thus (fg)m=f(gm).

 

[fresh_42]

To be honest, I have difficulties to understand your difficulties or to recognize any different definitions.

A group action of a group $G$ on a set $X$ (= group representation on $X$) is given by a group homomorphism $\varphi\, : \,G \longrightarrow \operatorname{Aut}(X)$. That's it.

Now to the details. Let $g,h,1 \in G$ and $x,y,v,w \in X\; , \;c \in \mathbb{F}$ where $\mathbb{F}$ is a field, e.g. real or complex numbers.

If authors speak of a group action, the notation of the homomorphism is usually simply dropped, that is they write $\varphi(g)(x) =: g.x \,.$ This is only a convention, as often the way the action is defined, namely by $\varphi$ is a natural one; e.g. matrices and vectors. However, it is helpful to keep this in mind.

Now to the next point: group homomorphism. This simply means $\varphi(g\cdot h)=\varphi(g)\cdot \varphi(h)\,.$ resp. $\varphi(g\cdot h)(x)=\varphi(g)(\varphi(h)(x))$ or short $(gh).x=g.(h.x)\,.$ Especially we have $1.x=\varphi(1)(x)=\operatorname{id}_X(x)=x$ and $\varphi(g \cdot g^{-1})(x)=(g.(g^{-1}.x)=(g\cdot g^{-1}).x=1.x=x\,.$

What's left is the explanation of $\operatorname{Aut}(X)$. This is simply the group of bijective transformations of $X\,.$ If $X$ carries an additional structure (group, vector space, etc.), then it is required, that the operation respects this structure. Therefore the abbreviation $\operatorname{Aut}$ for automorphism and not simply $\operatorname{Bij}$ for bijections. Now what does this mean? Say we have a binary operation $\circ$ on the set $X\,,$ then it is required that $g.(x \circ y) = \varphi(g)(x\circ y) = \varphi(g)(x)\circ \varphi(g)(y)=g.x \circ g.y\,.$

In case $X=V$ is a vector space and $G$ a group of matrices operating on this vector space, then we require $g.(v+w)=g.v+g.w$ and $g.(c\cdot v)=c\cdot g.v$ which certainly holds for matrices $g$, and in this case $\operatorname{Aut}(X)=\operatorname{GL}(V)\,.$ This is by far the most important example of an operation, but not the most general one, as $X$ doesn't have to carry an additional structure. A set $X$ will do. In group theory, $X$ is often the group itself or another group.

 

[Martin T]

Thanks your explanation was so clear , the reason of the post is If there exists the different between the action of a group and the action of a element g of a group G for example. The definition of ARTIN's book was nothing precise.

 

[fresh_42]

Thanks your explanation was so clear , the reason of the post is If there exists the different between the action of a group and the action of a element g of a group G for example. The definition of ARTIN's book was nothing precise.

If you have a group $G$ acting on a set $X$, then the the action of a single group element $g \in G$ is called the orbit of $g: \,g.X=\{\,y\in X\,|\,y=g.x \text{ for some }x\in X\,\}\,.$

 

[Martin T]

It is done. Really Thanks a lot.