Proving Isomorphism of (ZxZxZ)/<(2,4,8)> to (Z(index 2)xZxZ)

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However, this may require more advanced mathematical knowledge. Overall, your approach of defining a homomorphism and proving its properties is a valid and systematic way to solve this problem.
  • #1
I'm trying to solve (find a group that is ismorphic to) (ZxZxZ)/<(2,4,8)>.
(1,2,4)+<(2,4,8)> must be of order 2 in the factor group. (0,1,1)+<(2,4,8)> and (0,0,1)+<(2,4,8)> generates infinite cyclic subgroups of the factor group. So it would be reasonable to presume that (ZxZxZ)/<(2,4,8)> is isomorphic to (Z(index 2)xZxZ). To prove the presumption I have to define a homomorphism, h, mapping (ZxZxZ) onto (Z(index 2)xZxZ) having kernel <(2,4,8)>. The following properties h(1,2,4)=(1,0,0) and h(0,1,1)=(0,1,1) and h(0,0,1)=(0,0,1) and h((k,l,m)+(n,o,p)+(q,r,s))=h(k,l,m)+h(n,o,p)+h(q,r,s) must hold. h must also be onto. The last step is to prove that the kernel of h is contained in <(2,4,8)> and that <(2,4,8)> is contained in the kernel. All of this would prove that the factor group is isomorphic to (Z(index 2)xZxZ).

Is the reasoning above correct? Is there an easier way?

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  • #2
Yes, the reasoning you have provided is correct. There is not necessarily an easier way to solve this problem but there are some alternate methods that could be used. For example, you could use the Smith Normal Form to find an isomorphism between (ZxZxZ)/<(2,4,8)> and (Z(index 2)xZxZ).
  • #3

I would say that your reasoning and approach are correct. You have correctly identified the necessary steps to prove the isomorphism between (ZxZxZ)/<(2,4,8)> and (Z(index 2)xZxZ). It may be possible to find an easier or more efficient way, but your method is a valid and logical way to approach the problem. Keep in mind that in mathematics, there may be multiple ways to solve a problem, so if you do find a different approach that works, that is also valid. Good luck with your proof!

1. What is the definition of isomorphism?

Isomorphism is a mathematical concept that refers to a one-to-one correspondence between two mathematical structures or objects. In other words, two structures are isomorphic if there is a way to match every element of one structure to an element of the other structure in a way that preserves the underlying structure.

2. How do you prove isomorphism?

To prove isomorphism between two structures, you must show that there is a function that maps elements from one structure to the other while preserving the structure. This function must be both one-to-one and onto, meaning that every element in the first structure is mapped to a unique element in the second structure, and every element in the second structure has a corresponding element in the first structure.

3. What is the significance of the symbol /< in the notation (ZxZxZ)/<(2,4,8)>?

The symbol /< in the notation (ZxZxZ)/<(2,4,8)> represents an equivalence relation, indicating that certain elements in the structure have been identified as equivalent. In this case, the elements (2,4,8) have been identified as equivalent, and therefore the structure (ZxZxZ)/<(2,4,8)> contains elements that are equivalent to (2,4,8) removed.

4. How does the structure (ZxZxZ)/<(2,4,8)> differ from (Z(index 2)xZxZ)?

The structure (ZxZxZ)/<(2,4,8)> is a quotient structure of the original structure (ZxZxZ), meaning that certain elements have been identified as equivalent and removed. On the other hand, (Z(index 2)xZxZ) is a direct product of the structures Z(index 2), Z, and Z, meaning that it contains all possible combinations of elements from these structures.

5. Can you provide an example of an isomorphism between (ZxZxZ)/<(2,4,8)> and (Z(index 2)xZxZ)?

Yes, an example of an isomorphism between these two structures is the function f: (ZxZxZ)/<(2,4,8)> → (Z(index 2)xZxZ), where f(a,b,c) = (a mod 2, b, c). This function maps every element in (ZxZxZ)/<(2,4,8)> to an element in (Z(index 2)xZxZ) while preserving the structure, making it a one-to-one and onto function, and thus proving the isomorphism between the two structures.

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