there are two ways to look at a group action.

1)as a "multiplication" GxA-->A given by (g,a)-->g.a (also written as g(a)).

2) as a homomorphism φ from G--->Sym(A) (the set of bijections on A, so each element of g maps to a permutation of A).

now, the kernel as per the first way of looking at a group action is the set {g in G: g.a = a, for all a in A}.

the kernel in the second way of looking at a group action is the set {g in G : , the identity function on A}

we want to show these are the same thing. so suppose g.a = a, for all a in A. then

for all a in A, that is , so .

on the other hand, suppose . then g.a = = a, so that g is in the set {g in G: g.a = a, for all a in A}.

the whole point of this exercise is to show you the two definitions of a group action "really are the same".