Tour:Homomorphism of groups

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This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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WHAT YOU NEED TO DO:
  • Read, and understand, the definition of homomorphism of groups.
  • Try proving that a composite of homomorphisms is a homomorphism and the inverse of a bijective homomorphism is a homomorphism.
  • Read, and understand, the definitions of four special kinds of homomorphisms. You've already seen one of these.

Definition

Textbook definition (with symbols)

Let G and H be groups. Then a map \varphi: G \to H is termed a homomorphism of groups if \varphi satisfies the following condition:

\varphi(ab) = \varphi(a) \varphi(b) for all a, b in G

Universal algebraic definition (with symbols)

Let G and H be groups. Then a map \varphi:G \to H is termed a homomorphism of groups if \varphi satisfies all the following conditions:

  • \varphi(ab) = \varphi(a) \varphi(b) for all a, b in G
  • \varphi(e) = e
  • \varphi(a^{-1}) = (\varphi(a))^{-1}

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.

Facts

Composition

If \alpha: G \to H and \beta: H \to K are homomorphisms, then the composite mapping \beta.\alpha is a homomorphism from G to K. This follows directly from either definition.

Inverse

If a homomorphism is bijective, then its set-theoretic inverse map is also a homomorphism.

Related terms

Endomorphism

Further information: endomorphism

An endomorphism of a group is a homomorphism from the group to itself. Note that every group always has the following two endomorphisms:

  • The trivial endomorphism that sends every element of the group to the identity element
  • The identity map that sends every element to itself

The endomorphisms of a group form a monoid, termed the endomorphism monoid, under composition.

Isomorphism

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.

Automorphism

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.

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Group action| UP: Introduction five (beginners)| NEXT: Equivalence of definitions of homomorphism
Expected time for this page: 20 minutes
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part