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.

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## 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(n_{1}+n_{2})=2(n_{1}+n_{2}) = 2n_{1}+2n_{2} = θ(n_{1})+θ(n_{2}) ∀ n_{1}, n_{2} ∈ 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(n_{1})=θ(n_{2}) ⇔2n_{1}=2n_{2} = n_{1}=n_{2}.

**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 HomomorphismFirst 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:**