Let a group homomorphism. Then the kernel of is a normal subgroup of , and the image of is a subgroup of . Also, for all and for all . In particular, taking we have for all , and taking we have , where and are the identity elements of and , respectively.
Some special homomorphisms have special names. If the homomorphism is injective, we say that is a monomorphism, and if is surjective we call it an epimorphism. When is both injective and surjective (that is, bijective) we call it an isomorphism. In the latter case we also say that and are isomorphic, meaning they are basically the same group (have the same structure). A homomorphism from on itself is called an endomorphism, and if it is bijective then it is called an automorphism.
|Date of creation||2013-05-17 17:53:21|
|Last modified on||2013-05-17 17:53:21|
|Last modified by||unlord (1)|
|Synonym||homomorphism of groups|
|Defines||epimorphism of groups|
|Defines||monomorphism of groups|
|Defines||automorphism of a group|