Subgroups of a group of even order

#group_theory

Let $G$ be a group of order $2 n$ for some positive integer $n > 1$ .
(a) Prove there exists a subgroup $K$ of $G$ of order $2$ .
(b) Suppose $K$ in (a) is a normal subgroup. Prove that $K$ is contained in the center $Z (G)$ . (Recall $Z (G) = {a \in G ∣ a b = b a for all b \in G}$ .)

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Let $G$ be a group of order $2n$ for some positive integer $n > 1$.
\begin{enumerate}[label=\alph*)]
	\item Prove there exists a subgroup $K$ of $G$ of order $2$.
	\item Suppose $K$ in (a) is a \underline{normal} subgroup. Prove that $K$ is contained in the center $\operatorname{Z}(G)$. (Recall $\operatorname{Z}(G)=\{a\in G\mid ab=ba\text{ for all }b\in G\}$.)
\end{enumerate}