SOLVE: Isomorphism Problem for Z252 X Z294 and Z42 X Z1764

[phyalan]

Homework Statement

is Z₂₅₂ X Z₂₉₄ isomorphic to Z₄₂ X Z₁₇₆₄? Explain.

Homework Equations

The Attempt at a Solution

I checked that the highest order of the element in both group are 1764, but don't really know how to justify if there is an isomorphism...Can anyone give me some hints?

 

[micromass]

Hi phyalan!

Let's first write your thingies in a more canonical form. For this I want you to take the prime decomposition of 252, 294, 42, 1764 and apply the following formula:

$$\mathbb{Z}_{ab}\cong \mathbb{Z}_a\times \mathbb{Z}_b$$

if gcd(a,b)=1.

For example, you could write

$$\mathbb{Z}_{84}\cong \mathbb{Z}_4\times\mathbb{Z}_3\times \mathbb{Z}_7$$.

Try to do this with your groups...

 

[phyalan]

So is
$$\mathbb{Z}_a\times \mathbb{Z}_b\cong \mathbb{Z}_b\times \mathbb{Z}_a$$ ?

if that's ok, then i m done.
Anyway, thank you micromass =)

 

[micromass]

Yes, that's true!