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