- #1
Martin T
- 12
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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).
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).