TITLE: How to prove group generated $\langle a \rangle$ by element $a$ is normal subgroup of G iff a is the center of G?
QUESTION [6 upvotes]: Problem: Let $G$ denote a group, and $H$ a subgroup of $G$.
Suppose $a$ to be an element of $ G$ of order 2. 

Prove that $\langle a \rangle$ is a normal subgroup of $G$ iff  a is
  in the center of G.

My Work and thoughts: 
Let's suppose $a\in G$ has order 2.
This implies $a^2=e$
Further suppose $\langle a \rangle$ is normal subgroup of $G$.
Define $N_G(H)= \{a\in G|aH a^-1\}= \langle a \rangle$ as normalizer. $(h\in H)$
NTS $aha^{-1 }= ah=ha$ because $C_G(H)=\{a\in G|ah=ha\}$.
Is this the right idea? Of course I'm starting with $P \Rightarrow Q$ direction then I'll do $Q  \Rightarrow P$. I'm stuck. I'm not sure how to continue.

REPLY [6 votes]: If $\langle a\rangle$ is a normal subgroup of order 2, then for all $g\in G$, either $gag^{-1}=a$ or $gag^{-1}=e$. The latter is impossible because that would imply that $a=e$. So $a$ is in the center.