Group actions

poorna · #1 $Old$ July 4th, 2009, 08:00 AM

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?

TheAbstractionist · #2 $Old$ July 4th, 2009, 06:12 PM

$(1)\ \implies\ (2)$

Consider the map $\phi:G\times X\to X,\ \phi((g,x))=\psi(g)(x).$

$(2)\ \implies\ (1)$

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