
Group actions
Hey, We can define a group action in two ways:
1) A group G is said to act on a set X, if there exists a group homomorphism ψ : G > S(X) [the symmetric group of X].
2)Equivalently, a group G acts on a set X, if there is a map from G x X > X which assigns to each ordered pair <g,x>>g.x, such that :
For all x Є X, e.x = x
(h.g).(x) = h.(g.x), where h, g Є G, and x Є X
How are these two definitions equivalent?

Consider the map
For each define by for all Then for all is a bijection from to itself) and the required homomorphism is defined by for all for all