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:
- 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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