Let $G$ and $K$ be groups with identity elements $e_G$ and $e_K$, respectively, and let $\varphi: G \to K$ be a [group homomorphism](/page/Group%20Homomorphism).

1. If $H \le G$, then $\varphi(H) \le K$. 2. If $L \le K$, then $\varphi^{-1}(L) \le G$, where $\varphi^{-1}(L) := \{g \in G : \varphi(g) \in L\}$ is the set-theoretic preimage. 3. If $L \trianglelefteq K$, then $\varphi^{-1}(L) \trianglelefteq G$. 4. If $\varphi$ is surjective and $N \trianglelefteq G$, then $\varphi(N) \trianglelefteq K$.