
abstract algebra help
Given $\displaystyle K$ is a normal subgroup of a group $\displaystyle G$. The map $\displaystyle \psi : G$ > $\displaystyle G/K $ by $\displaystyle \psi(g) = \phi(g)K$ for $\displaystyle g \in G$ is a homomorphism and $\displaystyle \phi \in Aut(G)$.
Show $\displaystyle \psi(K) = (\phi(K)K)/K$.

Since $\displaystyle K$ is a subgroup of $\displaystyle G$, $\displaystyle \psi(K)$ is a subgroup of $\displaystyle G/K$. So for every $\displaystyle x \in \psi(K), x \in G/K$, and then $\displaystyle x = gK$ for some $\displaystyle g \in G$.
I need to show that $\displaystyle \psi(K)$ and $\displaystyle (\phi(K)K)/K$ are subsets of each other, but I'm stuck in here, did I start this right?