Isomorphism between $\mathbb{Z}_{2} \times \{0\}$ and $\langle (0,2) \rangle$

Question

I'm new to group theory and I'm learning about isomorphisms and quotient groups.

I'm stuck trying to show that that $\mathbb{Z}_{2} \times \{0\} \cong \langle (0,2) \rangle$.

I understand that two groups being isomorphic is defined as there existing a homomorphism between them that is bijective.

So my thought process was to try and define some function that bijectively maps $\{(0,0),(1,0)\}$ to $\{(0,0),(0,2)\}$.

I believe that this is probably a lot more simple than I'm making it out to be, but would like to check to make sure that I'm on the right path.

Also, please tell me if there are any formatting mistakes in this post and I will try to correct them.

Answer 1

Both groups are made of two elements -a trivial one (identity) and another element of order 2. There is only one way to create an isomorpism from one group to the other. Map the identity to the indentity and the element of order 2 to the element of order two of the other group.

Note that those two groups are basically the same - up to changing the 'names' of the elements, they behave in the exact same way. That's exactly the meaning of an isomorphism.