Homomorphism of groups
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Textbook definition (with symbols)
Let and be groups. Then a map is termed a homomorphism of groups if satisfies the following condition:
for all in
Universal algebraic definition (with symbols)
Let and be groups. Then a map is termed a homomorphism of groups if satisfies all the following conditions:
- for all in
Equivalence of definitions
For full proof, refer: Equivalence of definitions of homomorphism of groups
The textbook definition and universal algebraic definition of homomorphism of groups are equivalent. In other words, for a map between groups to be a homomorphism of groups, it suffices to check that it preserves the binary operation.
If and are homomorphisms, then the composite mapping is a homomorphism from to . This follows directly from either definition.
If a homomorphism is bijective, then its set-theoretic inverse map is also a homomorphism.
Further information: endomorphism
- The trivial endomorphism that sends every element of the group to the identity element
- The identity map that sends every element to itself
Further information: isomorphism of groups
An isomorphism of groups is a bijective homomorphism from one to the other. Equivalently it is a homomorphism for which an inverse map exists, which is also a homomorphism.
If there exists an isomorphism between two groups, they are termed isomorphic groups. Isomorphic groups are equivalent with respect to all group-theoretic constructions. In fact, the isomorphism gives the equivalence.
Further information: automorphism of a group
An automorphism of a group is an isomorphism of the group with itself. Equivalently, it is a bijective endomorphism from the group to itself.
Note that every injective endomorphism may not be an automorphism. Similarly, any surjective endomorphism may not be an automorphism.
Automorphisms of groups can be viewed as symmetries of the group structure. The collection of automorphisms of a group forms a group under composition and this is termed the automorphism group of the given group.
Kernel and image
The kernel of a homomorphism is defined as the set of elements that get mapped to the identity element in the image. It is a basic result of group theory that a subgroup of a group can be realized as the kernel of a homomorphism of a groups if and only if it is a normal subgroup For full proof, refer: Normal subgroup equals kernel of homomorphism
The image of a homomorphism is the set of elements that occur as images of elements. It turns out that the image of any homomorphism is a subgroup of the group on the right. More generally, in fact, the image of any subgroup of a group on the left is a subgroup of the group on the right.For full proof, refer: Homomorphic image of subgroup is subgroup
Further, it is also true that the inverse image of any subgroup of the group on the right is a subgroup of the group on the left.For full proof, refer: Inverse image of subgroup is subgroup
Expressibility as composite of surjective and injective homomorphism
Given any homomorphism of groups, suppose . Then, we can view as the composite of a surjective homomorphism (viz a quotient map) from to and an injective homomorphism (viz a subgroup inclusion) from to . Moreover, this expression (as composite of a surjective and injective homomorphism) is essentially unique.
Commutes with arbitrary words
Further information: homomorphism commutes with word maps
Suppose is a word in the letters (these are just formal symbols). Suppose is a homomorphism of groups. Then, commutes with , i.e.:
where the on the left is the word map in (i.e., it evaluates the word for a tuple of values of the letters in and the on the right is the word map in .
Specification on a generating set
To specify a homomorphism from one group to another, it suffices to specify the values that the homomorphism takes on a generating set for the group.
This is because the value that the homomorphism takes on any word in terms of certain elements is determines by the values taken on those elements.
Some easy corollaries:
- If a homomorphism of a group is known on two subgroups of the group, then it is also known on the subgroup generated by them.