Group Isomorphism: Definition, Properties, Examples

An isomorphism of groups is a special kind of group homomorphisms. It preserves every structure of groups. In this article, we will learn about isomorphism between groups, related theorems, and applications.

Definition of Isomorphism

A map Φ: (G, 0) → (G′, *) between two groups is called an isomorphism if the following conditions are satisfied:

  • Φ is a group homomorphism, that is, Φ(ab)=Φ(a)Φ(b) ∀ a, b ∈ G.
  • Φ is one-to-one.
  • Φ is onto.

A bijective group homomorphism between groups is called an isomorphism.

For example, the identity map i: Z → Z defined by i(n)=n ∀ n ∈ Z is an example of an isomorphism. Below are a few more examples of isomorphism of groups.

The map Φ: (Z5, +) → (Z5, +) defined by Φ($\bar{x}$)=3$\bar{x}$ ∀ $\bar{x}$ ∈ Z5 is an example of group isomorphism.

The map Φ: (Z, +) → (2Z, +) defined by Φ(n)=2n ∀ n ∈ Z is an isomorphism.

Properties of Isomorphism

Property 1: If Φ: (G, 0) → (G′, *) is a group isomorphism, then the kernel of the map is trivial, that is, ker(Φ)={eG}.

Proof: For a proof, visit the page: Injectivity criteria for homomorphism.

Property 2: If Φ: (G, 0) → (G′, *) is a group isomorphism, then we have:

  1. order of a = order of Φ(a) ∀ a ∈ G
  2. Both G and G′ have the same cardinality.

Property 3: Let Φ: (G, 0) → (G′, *) be a group isomorphism. Then the following are true:

  1. G is abelian if and only if G′ is abelian
  2. G is cyclic if and only if G′ is cyclic

Remark:

  • We see that both abelian and cyclic properties are preserved by a group isomorphism.
  • If a is a generator of G, then Φ(a) is a generator of G′.

Property 4: If Φ: (G, 0) → (G′, *) is a group isomorphism, then the inverse map Φ-1: (G′, *) → (G, 0) is also an isomorphism.

Property 5: The composition of two isomorphisms is an isomorphism.

Non Isomorphic Groups

Example 1: The groups (Z, +) and (Q, +) are not isomorphic.

Solution:

We know that (Z, +) is a cyclic group whereas (Q, +) is a non-cyclic group, see the page on cyclic groups. As the cyclic property is preserved by an isomorphism, we conclude that both the additive groups Z and Q are not isomorphic.

Example 2: The groups (Q, +) and (R, +) are not isomorphic.

Solution:

If there is an isomorphism between the additive groups Q and R, then they must have the same cardinality. But one knows that both Q and R have different cardinalities. So (Q, +) and (R, +) cannot be isomorphic.

Example 3: The groups (Q, +) and (Q+, .) are not isomorphic.

Solution:

Example 4: The groups (R×, .) and (R, +) are not isomorphic.

Solution: