
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?

$\displaystyle (1)\ \implies\ (2)$
Consider the map $\displaystyle \phi:G\times X\to X,\ \phi((g,x))=\psi(g)(x).$
$\displaystyle (2)\ \implies\ (1)$
For each $\displaystyle g\in G,$ define $\displaystyle \psi_g:X\to X$ by $\displaystyle \psi_g(x)=g\cdot x$ for all $\displaystyle x\in X.$ Then $\displaystyle \psi_g\in S(X)$ for all $\displaystyle g\in G$ $\displaystyle (\psi_g$ is a bijection from $\displaystyle X$ to itself) and the required homomorphism $\displaystyle \psi:G\to S(X)$ is defined by $\displaystyle \psi(g)=\psi_g$ for all $\displaystyle g\in G$ $\displaystyle (\psi_{hg}=\psi_h\psi_g$ for all $\displaystyle h,g\in G).$