Definition for the center of a Group:
The center $Z(G)$ of a group $G$ is the subset of elements in $G$ that commute with every other element of $G$.
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 $ab \in G$?