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.
Table of Contents
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:
