Kernel of a group action and Sa

Dummit and Foote Section 1.7 Exercise 5 reads as follows:

Prove that the kernel of an action of the group G on a set A is the same as the kernel of the corresponding permutation representation G $\displaystyle \longrightarrow$ $\displaystyle S_A$

I am having real trouble with this - can anyone help me?

By the way - what is the meaning of "the corresponding permutation representation$\displaystyle \longrightarrow$ $\displaystyle S_A$" ???

I would be really grateful for some assistance.

Peter

Re: Kernel of a group action and Sa

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 $\displaystyle \sigma_g$ 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 : $\displaystyle \sigma_g = id_A$, 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

$\displaystyle \sigma_g(a) = a$ for all a in A, that is $\displaystyle \phi(g) = \sigma_g = id_A$, so $\displaystyle g \in ker(\phi)$.

on the other hand, suppose $\displaystyle g \in ker(\phi)$. then g.a = $\displaystyle \sigma_g(a) = \phi(g)(a) = id_A(a)$ = 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".