Definition for the center of a Group:

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> The center $Z(G)$ of a group $G$ is the subset of elements in $G$ that commute with every other element of $G$.
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**Theorem:** *The center of a group $G$ is a subgroup of $G$.*

The author begins the proof by stating the it is obvious that 

$$e \in Z(G)$$ (why? I understand that the identity element is necessarily in the group G but is it necessarily in a subset?)

The author proceeds then to state 

$$(ab)x = a(bx) = (ax)b = (xa)b = x(ab) \qquad\forall x \in G$$ and therefore 
$$ab \in G$$

How does the commutativity property convince the reader that $ab \in G$?