In general, a group homomorphism, sends subgroups of to subgroups of . Also, the preimage of any subgroup of is a subgroup of . We call the preimage of the trivial group in the kernel of the homomorphism and denote it by . As it turns out, the kernel is always normal and the image of , is always isomorphic to (the first isomorphism theorem). In fact, this correspondence is a bijection between the set of all quotient groups of , , and the set of all homomorphic images of (up to isomorphism). It is also easy to see that the kernel of the quotient map, , is itself, so the normal subgroups are precisely the kernels of homomorphisms with domain .
In general a group homomorphism sends subgroups of to subgroups of  Also the preimage of any subgroup of is a subgroup of  We call the preimage of the trivial group in the kernel of the homomorphism and denote it by  As it turns out the kernel is always normal and the image of  is always isomorphic to (the first isomorphism theorem) In fact this correspondence is a bijection between the set of all quotient groups of   and the set of all homomorphic images of (up to isomorphism) It is also easy to see that the kernel of the quotient map  is itself so the normal subgroups are precisely the kernels of homomorphisms with domain 