Finding Isomorphisms for Group G x Z & G x G

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In summary, the conversation discussed finding an isomorphism from G x Z to G and from G x G to G. The group G was defined as infinite sequences of integers that combine termwise. The group G was also shown to be isomorphic to the infinite Cartesian product ZN, which can be used to demonstrate the isomorphism of G x Z to G. The use of arithmetic rules for sets was also mentioned as a helpful tool for these calculations.
  • #1
Obraz35
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I have the group G whose elements are infinite sequences of integers (a1, a2, ...). These sequences combine termwise as such:
(a1, a2,...)(b1, b2,...) = (a1+b2, a2+b2,...)

I would like to find an isomorphism from G x Z (the direct product of G and the integers) to G as well as an isomorphism from G x G to G.

So far, I have found several homomorphisms for both of these but all of them lack the injective property so fail to be isomorphisms. What sorts of functions can I construct that are isomorphisms to G for these two groups?

Thanks.
 
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  • #2
In the arithmetic of Abelian groups, your group G is isomorphic to the infinite Cartesian product ZN... (actually, it's probably equal to, not just isomorphic to, but that doesn't matter)
 
  • #3
Okay. I see that, but I guess I'm not sure how to use that fact to show that G x Z is isomorphic to G. Should I try to show that G x Z is also isomorphic to the infinite direct product?
 
  • #4
That would work. Calculations like this are one of the reasons we learn how to do arithmetic with sets!

(I'm assuming you know the appropriate arithmetic rules, such as [itex](A \times B)^C \cong A^C \times B^C[/itex])
 
  • #5
Yes, thanks very much.
 

1. What is an isomorphism in the context of group theory?

An isomorphism is a bijective function that preserves the structure of a group. This means that the operation and identity elements are preserved, and the function maps each element in one group to a unique element in another group.

2. Why is finding isomorphisms for G x Z and G x G important?

Isomorphisms provide a way to show that two groups have the same structure, even if they may look different. This can help in understanding and comparing different groups, and can also be used to prove certain properties or theorems.

3. How do you find an isomorphism for G x Z?

One approach is to use the Fundamental Theorem of Finite Abelian Groups, which states that every finite abelian group can be uniquely decomposed as a direct product of cyclic groups. By decomposing G and Z into cyclic groups and finding a bijection between their elements, we can construct an isomorphism for G x Z.

4. Can an isomorphism exist between G x Z and G x G?

Yes, an isomorphism can exist between these two groups if they have the same order. However, it is not always guaranteed to exist, as the structure of G x Z and G x G may differ even if they have the same order.

5. Are there any other methods for finding isomorphisms for G x Z and G x G?

There are various techniques and algorithms that can be used to find isomorphisms between groups, such as the Cayley table method or the subgroup method. However, the most efficient method will depend on the specific groups being considered.

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