# Isomorphism: Klein 4 and Z2 x Z2

• Sep 11th 2011, 02:28 AM
Bernhard
Isomorphism: Klein 4 and Z2 x Z2
Can anyone help with the following problem

Let V be the Klein 4 group

Show that V \$\displaystyle \cong \$ \$\displaystyle \mathbb{Z}_2}\$ x \$\displaystyle \mathbb{Z}_2}\$

At the moment I am testing various possibilities - but there must be a better approach than just randomly attempting to construct an isomorphism!

Be most grateful for help.

Peter
• Sep 11th 2011, 04:36 AM
TheChaz
Re: Isomorphism: Klein 4 and Z2 x Z2
Uh... You have four elements: 1, a, b, ab
They map to four elements: (0, 0), (0, 1), (1,0), (1,1)
What is random??
Identity maps to identity. Then every other element has order two...
• Sep 11th 2011, 05:13 AM
Bernhard
Re: Isomorphism: Klein 4 and Z2 x Z2
Can you explain how you are using the fact that every other element has order 2?

Are you saying that therefore you must get an isomorphism however you match the other 3 elements of each group?

Peter
• Sep 11th 2011, 05:20 AM
TheChaz
Re: Isomorphism: Klein 4 and Z2 x Z2
It matters to which element ab maps.
Here is your isomorphism from V to Z2 x Z2:
f(1) = (0,0)
f(a) = (1,0)
f(b) = (0,1)
f(ab) = (1,1)
Isn't the defining feature of the Klein four group (if not that it's isomorphic to Z2xZ2!) its multiplication table? You will have the same (additive) table ...
• Sep 11th 2011, 05:36 AM
Bernhard
Re: Isomorphism: Klein 4 and Z2 x Z2
Thanks

But where did your reasoning with the orders of the elements come into it?
• Sep 11th 2011, 08:10 AM
TheChaz
Re: Isomorphism: Klein 4 and Z2 x Z2
Quote:

Originally Posted by Bernhard
Thanks

But where did your reasoning with the orders of the elements come into it?

It didn't, explicitly.
I'm guessing as to how the K group was defined for you.
This might be useful;
Group where every element is order 2 - Mathematics - Stack Exchange