Theorem 14.1

Let $\phi: G \rightarrow G'$ be a group homomorphism with kernel $H$. Then the cosets of H form a factor group, $G / H$, where $(aH)(bH) = (ab)H$. Also, the map $\mu: G / H \rightarrow \phi[G]$ defined by $\mu(aH) = \phi(a)$ is an isomorphism. Both coset multiplication and $\mu$ are well defined, independent of the choices $a$ and $b$ from the cosets.