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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).