Group action on set Let $A=\{a,b,c,d\}$ be a set consisting of 4 distinct elements. In this question, a group action $s:\mathbb{Z}_4\to S_A$ is considered. Here $\mathbb{Z} _4=\{0,1,2,3\}$ while the group operation in the group $(\mathbb{Z}_4, +_4)$ is addition modulo 4. 

Define $s_2 = s(2)$. Show that either $s_2 = e_A$ or that the permutation $s_2$ has order 2. 

I have an idea how to solve this but I'm missing some fundamental  knowledge. My problem is understanding what $s_2 = s(2)$ actually means. Is 2 the element 2 from G? And if so does it make sense to ask what s(2) equals? 
Sorry if my question isn't that clear or doesn't make sense.
 A: $s$ is a map $s:\mathbb{Z}_4\to S_A$. To each element of $\mathbb{Z}_4$, it associates a permutation on $A$, i.e. a "shuffling of the elements" of $A$. In particular, $s(2)$ determines a certain way of shuffling these elements. Since $s(2)$ is a map $A\to A$, we may prefer to denote it $s_2$, such that we can write $s_2 (x)$ instead of the more awkward $s(2)(x)$.
Below is a proposed solution to the exercise.
As $s$ is a group homomorphism, it respects operations of each group, in the sense that
$$s(2+2)=(s(2))\circ (s(2))$$
where $\circ$ denotes map composition, since this is the group operation on $S_A$.
Writing $s_2$ instead of $s(2)$, and using that $2+2=0$ in $\mathbb{Z}_4$, this becomes
$$s(0)=s_2\circ s_2.$$
But $s$ being a group homomorphism, it must map the identity element of $\mathbb{Z}_4$ to the identity element of $S_A$, i.e. $s(0)=e_A$. Hence
$$s_2\circ s_2=e_A,$$
which precisely means that $s_2$ has order $2$, unless $s_2=e_A$ to begin with, in which case it has order 1.
A: $2$ is the element of $\mathbb Z_4=\{0,1,2,3\}$. Since $s$ is a function from $\mathbb Z_4$ to $S_A$, $s(2)$ is an element of $S_A$ -- which particular element it is depends on what $s$ is, of course.
In particular it is possible for $s(2)$ to be $e_A$, the identity permutation. For example this would be the case if $s$ maps every element of $\mathbb Z_4$ to $e_A$ (you can, and possibly should, check that this satisfies the conditions for being a group action).
