Showing Isomorphisms in Subgroups

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In summary, the conversation discusses the subset G of Z x Z where G = {(a,b)|a+b = 2k for some integer k}. The person is trying to show that G is a proper subgroup of Z x Z and determine if it is isomorphic to Z x Z. They have shown that it is a proper subgroup but are struggling with the isomorphism part. It is mentioned that Z x Z and G are both abelian groups and can be generated by two elements. A hint is given to find the generators by drawing Z x Z in a plane and circling the elements of G. The person initially thinks that (2,0), (0,2), and (1,1) are the generators for
  • #1
Obraz35
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Homework Statement


Let G be a subset of Z x Z (direct product) where G = {(a,b)|a+b = 2k for some integer k}.
I'd like to show that G is a proper subgroup of Z x Z and determine whether G is isomorphic to Z x Z.

I am pretty sure I have shown that it is a proper subgroup but the isomorphism part is what is giving me trouble. Here the cardinality of the two sets is equal and it seems reasonable to be able to find a bijection between them, but I am not sure on the details of showing this.
 
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  • #2
It may seem reasonable that if the cardinalities are equal there should be an isomorphism, but that's not necessarily true. In this case it is. Z x Z is an abelian group which can be generated by two elements. G is also an abelian group with two generators. Can you find them? Hint: draw Z x Z in the plane and circle the elements belonging to G. It's a lattice. You can get an isomorphism by mapping generators to generators.
 
  • #3
I see the visual representation, but I guess I am just not seeing how this gives you the generators. The only way I can think to generate G is with (2,0), (0,2) and (1,1). I'm also not sure how to map the rest of the elements that are not generators.
 
  • #4
(0,2)=(-1)*(2,0)+2*(1,1). You don't need three generators. Clearly, (a,b)=a*(1,0)+b*(0,1). Only two generators. Your G is not that much different.
 
  • #5
But don't the generators of G have to lie in G? Because 1+0 is odd.
 
  • #6
No, I meant (1,0) and (0,1) generate Z x Z. The point is that you can also find two elements that generate G.
 
  • #7
Okay, I see it now. Thanks.
 

1. What is an isomorphism in subgroups?

An isomorphism in subgroups is a type of mathematical mapping between two groups that preserves their structure and operations, while also maintaining a one-to-one correspondence between elements of the groups. This means that the groups have the same number of elements and the same group operations, but the elements may be represented differently in each group.

2. How do you prove that two subgroups are isomorphic?

To prove that two subgroups are isomorphic, you need to show that there exists a bijective homomorphism (a mapping that preserves the group structure) between the two subgroups. This can be done by demonstrating that the elements of one subgroup can be mapped to the elements of the other subgroup, while maintaining the group operation.

3. What are some common techniques for showing isomorphisms in subgroups?

One common technique for showing isomorphisms in subgroups is to use the first isomorphism theorem, which states that if there exists a homomorphism between two groups, then the quotient of the two groups is isomorphic to the image of the homomorphism. Another technique is to show that the subgroups have the same order (number of elements) and the same group operation.

4. Can two subgroups be isomorphic if they have different structures?

Yes, two subgroups can still be isomorphic even if they have different structures. This is because isomorphisms only require that the groups have the same number of elements and the same group operation, but the elements may be represented differently in each group.

5. Are all subgroups of a group isomorphic to each other?

No, not all subgroups of a group are isomorphic to each other. Isomorphisms between subgroups are only possible if the subgroups have the same order and group operation. However, a group can have multiple isomorphic subgroups.

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