Group Homomorphism: Definition, Examples, Properties

A group homomorphism is a map between two groups that preserves the algebraic structure of both groups. In this section, we will learn about group homomorphism, related theorems, and their applications.

Definition of Group Homomorphism

A map Φ: G → G′ between two groups  (G, 0) and (G′, *) is called a group homomorphism if the group operation is preserved in the following sense:

Φ(a$\circ$b)=Φ(a)*Φ(b) ∀ a,b ∈ G

Example of Group Homomorphism

The following is an example of a group homomorphism. The map θ: (Z, +) → (Z, +) defined by

θ(n)=2n ∀ n ∈ Z

is a group homomorphism, because

θ(n1+n2)=2(n1+n2) = 2n1+2n2 = θ(n1)+θ(n2) ∀ n1, n2 ∈ Z

One-to-One homomorphism:

A group homomorphism Φ: G → G′ is said to be one-to-one (or into) if the map Φ is one-to-one. In other words, Φ is one-to-one if the following holds:

a=b if and only if Φ(a)=Φ(b) where a, b ∈ G.

The above map θ is an example of into homomorphism as θ(n1)=θ(n2) ⇔2n1=2n2 = n1=n2.

Onto homomorphism:

A group homomorphism Φ: G → G′ is called onto (or surjective) if the map Φ is onto. That is, every element of G has a preimage under the map Φ. It means that for any g′ ∈ G′ we have some g ∈ G such that Φ(g)=g′.

The above map θ is an example of onto homomorphism. This is because for any even integer 2n ∈ Z we have n ∈ Z such that θ(n)=2n.

Properties of Group Homomorphism

  • A one-to-one group homomorphism is called a monomorphism.
  • An onto group homomorphism is called an epimorphism.
  • A group homomorphism is called an isomorphism if it is both one-to-one and onto.
  • An isomorphism from a group G onto itself is called an automorphism.
Also Read:

Order of a Group: The order of a group and of its elements are discussed here with formulas.

Abelian Group: Definition, Properties, Examples

Cyclic Group: The definition, properties, and related theorems on cyclic groups are discussed.

Kernel of a Homomorphism

First Isomorphism Theorem: Proof and Application

Theorems of Group Homomorphism

Let (G, 0) and (G′, *) be two groups and let Φ: G → G′ be a group homomorphism.

Theorem 1: Φ(eG) = eG′

That is, a group homomorphism maps identity to identity.

Proof:

We know that eG$\circ$eG = eG in G. This implies that Φ(eG$\circ$eG) = Φ(eG). Since Φ is a homomorphism, we have that

Φ(eG) * Φ(eG) = Φ(eG)

⇒ Φ(eG) * Φ(eG) = Φ(eG) * eG′

⇒ Φ(eG) = eG′ by the left cancellation law. proved.

Theorem 2: Φ(a-1) = {Φ(a)}-1 for all a ∈ G.

Proof:

For a ∈ G, we have a$\circ$a-1 = eG = a-1$\circ$a, where a-1 denotes the inverse of a.

⇒ Φ(a) * Φ(a-1) = Φ(eG) = Φ(a-1) * Φ(a) as Φ is a homomorphism.

⇒ Φ(a) * Φ(a-1) = eG′ = Φ(a-1) * Φ(a) by Theorem 1.

So by the definition of an inverse, we conclude that Φ(a-1) is the inverse of Φ(a). In other words,

Φ(a-1) = {Φ(a)}-1 proved.

Theorem 3: If a ∈ G and the order of a is finite, then the order of Φ(a) is a divisor of the order of a. In other words,

$\circ(a) \mid \circ(\phi(a))$

Proof: