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For a homomorphism f, the quotient G/ker(f) is isomorphic to im(f)
Proposition (Quotient by the kernel). Let be a group homomorphism. Then there exists a unique group isomorphism
such that for all . In particular,
as groups, where is the quotient group and is a subgroup of .
Context. This is the standard "hands-on" form of the first isomorphism theorem. It identifies precisely what information about is "lost" under : exactly the kernel.
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 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 .
Let be a group homomorphism. The kernel of is the subset