# Thread: Isomorphism from C^* to G/H

1. ## Isomorphism from C^* to G/H

If we have the group $G=(C^*,\times)$ and the subgroup $H=\{\pm1,\pm i\}$. Prove that $(C^*,\times)/H$ is isomorphic to $(C^*,\times)$.

As the cosets of H in G are $\{\pm a\pm ib\}\ \forall{a,b}\in\mathbb{R}$.
Define $\varphi:\mathbb{C}^\times\to\mathbb{C}^\times$ by $\varphi(z)=z^4$. Prove that $\varphi$ is a group homomorphism and apply the first isomorphism theorem.