> [!NOTE] **Definition** > > Let $G$ and $H$ be [[Groups|groups]] and let $\phi: G \to H$ be a [[Homomorphisms of groups|homomorphism]]. Then the *kernel* of $\phi$ is the [[Preimage (of set under a function)|preimage]] of $\{ 1_{H} \}$ under $\phi,$ denoted $\text{Ker }\phi = \{ g \in G \mid \phi(g) = 1_{H} \} .$ # Properties Note that [[Kernel of Homomorphisms of Groups is Normal Subgroup of Domain]].