group homomorphism
Let and be two groups. A group homomorphism is a
function such that
for all .
A composition of group homomorphisms is again a homomorphism
.
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.
Title | group homomorphism |
Canonical name | GroupHomomorphism |
Date of creation | 2013-05-17 17:53:21 |
Last modified on | 2013-05-17 17:53:21 |
Owner | yark (2760) |
Last modified by | unlord (1) |
Numerical id | 27 |
Author | yark (1) |
Entry type | Definition |
Classification | msc 20A05 |
Synonym | homomorphism |
Synonym | homomorphism of groups |
Related topic | Group |
Related topic | Kernel |
Related topic | Subgroup |
Related topic | TypesOfHomomorphisms |
Related topic | KernelOfAGroupHomomorphism |
Related topic | GroupActionsAndHomomorphisms |
Related topic | Endomorphism2 |
Related topic | GroupsOfRealNumbers |
Related topic | HomomorphicImageOfGroup |
Defines | epimorphism |
Defines | monomorphism |
Defines | automorphism |
Defines | endomorphism |
Defines | isomorphism |
Defines | isomorphic |
Defines | group epimorphism |
Defines | group monomorphism |
Defines | group automorphism |
Defines | group endomorphism |
Defines | group isomorphism |
Defines | epimorphism of groups |
Defines | monomorphism of groups |
Defines | automorphism of a group |
Defines | endom |