1. The problem asks which group is isomorphic to the Klein-4 group, denoted $V_4$, which is a group of order 4 with every element of order 2 except the identity. 2. Recall the Klein-4 group $V_4$ is isomorphic to the direct product $\mathbb{Z}_2 \times \mathbb{Z}_2$, which has four elements: $(0,0), (1,0), (0,1), (1,1)$, each of order 2 except the identity $(0,0)$. 3. Let's analyze each option: - a. $D_4$ is the dihedral group of order 8 (symmetries of a square), which has 8 elements, so it cannot be isomorphic to $V_4$ which has 4 elements. - b. $\mathbb{Z}_4$ is the cyclic group of order 4, generated by one element of order 4. In $V_4$, no element has order 4, so $\mathbb{Z}_4$ is not isomorphic to $V_4$. - c. $\mathbb{Z}_2 \times \mathbb{Z}_2$ is exactly the Klein-4 group by definition. - d. $\mathbb{Z}_{12} / \langle 3 \rangle$ is the quotient of $\mathbb{Z}_{12}$ by the subgroup generated by 3. The subgroup $\langle 3 \rangle$ has elements $\{0,3,6,9\}$, so the quotient has order $12/4=3$, which is not 4, so it cannot be isomorphic to $V_4$. 4. Therefore, the only group isomorphic to the Klein-4 group is $\mathbb{Z}_2 \times \mathbb{Z}_2$. **Final answer:** c. $\mathbb{Z}_2 \times \mathbb{Z}_2$