Let $\theta: G \rightarrow H$ be a homomorphism between two groups $G$ and $H$. Show that the image of $\theta, \theta(G)$, is a subgroup of $H$; show also that the kernel of $\theta, \operatorname{ker}(\theta)$, is a normal subgroup of $G$.

Show that $G / \operatorname{ker}(\theta)$ is isomorphic to $\theta(G)$.

Let $O(3)$ be the group of $3 \times 3$ real orthogonal matrices and let $S O(3) \subset O(3)$ be the set of orthogonal matrices with determinant 1 . Show that $S O(3)$ is a normal subgroup of $O(3)$ and that $O(3) / S O(3)$ is isomorphic to the cyclic group of order $2 .$

Give an example of a homomorphism from $O(3)$ to $S O(3)$ with kernel of order $2 .$