For any group G, define group action on a set A. There will be a permutation representation of that group action. I am kind of confused why the permutation representation can be used to reflect the structure of the group G. I have two questions.
1.In dummit and foote I did not see any proof of the existence of a homomorphism for any well defined group action with some permutation group action. Where can I find it?
2.Since there is a group homomorphism between G and some symmetry group $S_{n}$ defined by the action, is this homomorphism really isomorphism? (I thought this is essentially representing each group element of G as a permutation group element and this may also be used to reflect the structure of G.) Correct me if I am wrong.
This is proved in section 1.7 of Dummit and Foote's Abstract Algebra.
The permutation representation is a homomorphism, it is not an isomorphism in general. For example, the trivial group $G = \{e\}$ acts on $\{1, 2\}$ by the identity (i.e. $e\cdot 1 = 1$ and $e\cdot 2 = 2$), so the associated permutation representation is a group homomorphism $G \to S_2$, but clearly $G$ is not isomorphic to $S_2$.