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The kernel of a group homomorphism is a normal subgroup
Proposition (Kernel is normal). Let be a group homomorphism. Let be its kernel. Then is a normal subgroup of .
Context. Normal subgroups arise naturally as kernels; conversely, every normal subgroup is the kernel of a canonical map to a quotient group.
Let be groups. A group homomorphism is a function such that for all ,
(Here denotes the product in and the product in .)
Basic consequences of the defining identity include: and for all . Two fundamental associated subgroups are the kernel and the image of . These are tied together by the first isomorphism theorem.
Let be a group homomorphism. The kernel of is the subset
Let be a group and let be a subgroup. The subgroup is normal in (written ) if for every ,
Normality says that is stable under the conjugation action of on itself. Equivalently, is normal iff every left coset of equals the corresponding right coset, and this is exactly the hypothesis needed to form the quotient group .