Explain intuition about how group actions induce permutations on sets? Can some one please explain to me, in some kind of "English" or intuitive way, what is the meaning and purpose behind permutations of a set "induced by" an element of the group that's acting on the set?
Sometimes i think I understand it, because I guess a group action is a mapping...so is this action what is referred to as the permutation "induced" by G on A? Is the same permutation as the action? Or is there some other mapping involved that can be induced only if certain properties are true?
And, why do we care so much about this induced mapping? It seems like there is some special properties...like is it always an isomorphism which allows us to invoke certain theorems when solving problems? I keep seeing this tool used in proofs - "consider the homomorphism induced by the group action" - but I haven't fully understood what it means for sure and where it comes from, how we know that one exists, or why we would want to take note of it. 
Oh...and is this also what's called the "permutation representation?"
 A: In "pure" group theory, it is useful to know that every group is isomorphic to a subset of the set of all permutations of some underlying set. Note that I said isomorphic, which means that the group structure has been "preserved" upon passing to this set of permutations. Specifically, it has been replaced by function composition. 
The existence of this isomorphism is a "representation theorem": given an abstract object, it provides a concrete representation of that object, or at least a more concrete representation than the one you started with. It's fairly common in math (not even just in algebra) to introduce an abstract object, prove some representation theorem about it, and then use that representation to help you understand the abstract object and/or the corresponding concrete object.
In more "applied" group theory, like representation theory, the underlying set is of some interest per se, and then the notion of a group action is more useful.
A: You really seem to understand the idea already.
A group $G$ can give a collection of functions $X \to X$. For each $g$ in $G$ we get a function from $X$ to $X$.
We write this as $\alpha : G \times X \to X$. For any $g$ in $G$ and $x$ in $X$ we get something in $X$ called $\alpha(g,x)$.
For a fixed $g$ in $G$ we get a function $\alpha_g : X \to X$ given by $x \mapsto \alpha(g,x)$.
The nice property is that the group operation agrees with function composition: $$(\alpha_g \circ \alpha_h)(x) = \alpha(g,\alpha(h,x)) = \alpha(gh,x)= \alpha_{gh}(x) $$
As you say: we get a homomorphism between the group $G$ with its own operation, and a set of functions $X \to X$ with the operation of composition.
The most important idea is when $X$ is a vector space, say $V$. The group action becomes $\alpha : G \times V \to V$. We get a set of linear maps $V \to V$. We can then start looking at linear algebra things like determinant, eigenvalues, eigenvectors, trace, etc. There are different ways that a given group $G$ can act on a given vector space $V$. One of these actions is called a group representation. For each $g$ in $G$, we get a linear map $f_g : V \to V$. If we take coordinates on $V$, then for each $g$ in $G$, we get a matrix representing the linear map $f_g : V \to V$.
It turns out that the trace of $f_g$ is the important invariant. The trace of $f_g$ is the same for all $g$ in the same conjugacy class. Now we find ourselves in a position where the linear algebra tells us something about the group.
A: An action assigns to each element $g$ of the group and each element $a\in A$ an element $g(a)\in A.$ So for each fixed $g\in G$ we have a map $a \mapsto g(a)$ from $A$ to $A$ which is a bijection (the inverse map is given by $a \mapsto g^{-1}(a)$). This map is the induced permutation.
