# Definition Given a [[group homomorphism]] $f:G\rightarrow H$, with identities $e_G$ and $e_H$. The Kernel is defined to be the set: $ K \equiv \{g \in G | \, f(g) = e_H\} $ That is, the kernel $K$ is the set of all [[group elements]] in $G$ that get mapped to the identity in $H$ under the homomorphism $f$. $K$ is not only a [[subgroup]], but is actually a [[normal subgroup]] of $G$. If $f$ is [[Types of Maps|one-to-one]], then $K = \{e_g\}$ (since there's only one element in $G$ that maps to the identity in $H$, and [[Group Homomorphism#Properties|homomorphisms preserve the identity]]). If $f$ is *not* one-to-one, then the size of $K$ tells us how far away it is from being one-to-one. In the case of [[abelian group|abelian groups]], the discussion is virtually identical to that on [[vector spaces]], found [[Kernel of a Map|here]]. # Cosets Being a [[normal subgroup]], the left and right [[coset|cosets]] are equal. Consider the right coset $ Kg \equiv \{kg|k \in K\} \quad g \in G $ $Kg$ are *precisely* those elements of $G$ which get sent to $f(g) = h \in H$. Thus, if the *size* of $K$ tells us how far away $f$ is from being [[Types of Maps|injective]], then the elements of $K$ tell us exactly which elements of $G$ will map to a specific element $h \in H$.