Definition:Kernel (Abstract Algebra)

Let $$\phi: \left({S, \circ}\right) \to \left({T, *}\right)$$ be a group homomorphism.

The kernel of $$\phi$$ is the subset of the domain of $$S$$ defined as :

$$\mathrm {ker} \left({\phi}\right) = \left\{{x \in S: \phi \left({x}\right) = e_T}\right\}$$

That is, $$\mathrm {ker} \left({\phi}\right)$$ is the subset of $$S$$ that maps to the identity of $$T$$.