Let $G= \mathbb{Z}_4=\mathbb{Z}/4\mathbb{Z}$ and $H=\langle2\rangle$ Then $G= \mathbb{Z}_4 = \mathbb{Z}/4 \mathbb{Z} = \{0,1,2,3\}$.We know $\mathbb{Z}/4\mathbb{Z}$ is cyclic. Cayley Table of $G=\mathbb{Z}/4 \mathbb{Z}$ is:$\begin{array}{|c|c|c|c|c|}\hline\hline\textbf{+} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3}\\\hline \textbf{0} & 0 & 1 & 2 & 3 \\\hline \textbf{1} & 1 & 2 & 3 & 0 \\\hline \textbf{2} & 2& 3 & 0 & 1\\\hline\textbf{3} & 3 & 0 & 1 & 2 \\\hline \end{array}$   $\cong$   \(\begin{array}{|c|c|c|c|c|}\hline\hline\bf{+}…