I need help with this advanced algebra problem.
Let $G$ be a group. We call the set $C(G)= \{a \in G : ab=ba, \forall b \in G\}$ the center of $G$. Prove that:
(a) $C(G)$ is normal subgroup of $G$.
(b) $C(G)=G$ if and only if $G$ is abelian group.
(c) If $a$ is the only element in $G$, then $a\in C(G)$.