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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?
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\ \implies\ (2))
Consider the map )=\psi(g)(x).)
\ \implies\ (1))
For each 
define 
by =g\cdot x)
for all 
Then )
for all 
is a bijection from 
to itself) and the required homomorphism )
is defined by =\psi_g)
for all 
for all .)
