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The kernel of a group homomorphism is invariant under conjugation
Kernels are Normal Subgroups: Let be a group homomorphism between groups. Then the kernel is a normal subgroup of .
This fact is what makes quotient groups naturally arise from homomorphisms, and it underlies the first isomorphism theorem.
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 .
Let be a normal subgroup of a group . The quotient group is the set of (left) cosets equipped with the operation
First Isomorphism Theorem (Groups). Let and be groups, and let be a group homomorphism. Let be the kernel of and let be the image of , i.e.
Then is a normal subgroup of (see kernels are normal subgroups), and the induced map
is a well-defined isomorphism. In particular, if is surjective then .