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A homomorphism that kills a normal subgroup factors uniquely through the corresponding quotient group.
Let be a group, a normal subgroup, and the canonical map to the quotient group. If is a group homomorphism with , then there is a unique homomorphism such that
It is given by .
Thus composition with gives a bijection between homomorphisms and homomorphisms that send every element of to the identity of .
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
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
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.