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A group homomorphism induces an isomorphism from the quotient by its kernel to its image.
Let be a group homomorphism. Its kernel is a normal subgroup of , and induces a group isomorphism
In particular, if is surjective, then .
This is the basic “quotient equals image” principle. With , it can be expressed by the short exact sequence
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 subgroup
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 groups. A group isomorphism is a group homomorphism that is bijective as a bijective function.
If is an isomorphism, then its inverse function is also a group homomorphism, so gives a structure-preserving identification between and . One writes to denote that there exists a group isomorphism between them.
A sequence of group homomorphisms
is exact at if
where is the image and is the kernel. The sequence is exact if it is exact at every term.
A short exact sequence is an exact sequence of the form
which encodes that is injective, is surjective, and is a normal subgroup of , with the quotient group .