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.
Table of Contents
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:
- order of a = order of Φ(a) ∀ a ∈ G
- Both G and G′ have the same cardinality.
Property 3: Let Φ: (G, 0) → (G′, *) be a group isomorphism. Then the following are true:
- G is abelian if and only if G′ is abelian
- 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:
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