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(e_G) = e_G′

That is, a group homomorphism maps identity to identity.

Proof:

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

Φ(e_G) * Φ(e_G) = Φ(e_G)

⇒ Φ(e_G) * Φ(e_G) = Φ(e_G) * e_G′

⇒ Φ(e_G) = e_G′ 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 = e_G = a^-1$\circ$a, where a^-1 denotes the inverse of a.

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

⇒ Φ(a) * Φ(a^-1) = e_G′ = 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: