Kernel

In general, a kernel is a measure of the failure of a homomorphism to be injective.

In set theory, if $S$ and $T$ are sets, with $f$ a function mapping $S$ into $T$ , the kernel of $f$ is quotient set of $S$ under the equivalence relation $R(x,y)$ defined as " $f(x)=f(y)$ ".

In group theory, if $G$ and $H$ are groups, and $f : G \to H$ is a homomorphism of groups, the kernel of $f$ is the set of elements of $G$ that map to the identity of $H$ , i.e., the set $f^{-1}(e_{H})$ . The kernel is a normal subgroup of $G$ , and in fact, every normal subgroup of $G$ is the kernel of a homomorphism.

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Kernel