# group theory question

Let $A=\left\{x\ :\ x\in G\ \text{and}\ x\neq x^{-1}\right\}$ and $B=\left\{x\ :\ x\in G\ \text{and}\ x=x^{-1}\right\}$. We have $A\cap B=\emptyset$ and $G=A\cup B$ so $|G|=|A|+|B|$. Can you show that $|B|$ is even and that there exists an element $x\neq e$ lying in $B$?