1. Direct products and isomorphisms

By the Chinese Remainder Theorem,

if gcd(p,q) > 1, then ℤp x ℤq ≇ ℤpq

Does this imply that if ℤa x ℤb ≇ ℤc and ℤux ℤv ≇ ℤc, then ℤa x ℤb ≇ ℤux ℤv ?

Or is there something I'm missing?

Thanks!

2. Re: Direct products and isomorphisms

In general, $\mathbb Z_m\times\mathbb Z_n\cong\mathbb Z_{\mathrm{lcm}(m,n)}$.

3. Re: Direct products and isomorphisms

Yes, but what I wanted to check was if there are two different direct products both not isomorphic to the same ℤ group, does that imply that the two different direct products are also not isomorphic to each other?

4. Re: Direct products and isomorphisms

Hi,
First, I think Nehushtan's response is a little misleading. If m and n are not relatively prime, the direct product $\mathbb Z_m\times \mathbb Z_n$ is not cyclic.
Next, I really don't know what you're asking. The relation "not isomorphic" is never transitive.

5. Re: Direct products and isomorphisms

I've been delving into the theory of it and I think I'm starting to understand a bit more, I probably didn't phrase my original question very well. I'm trying to work out if a direct product of two ℤ groups is isomorphic to the direct product of two different ℤ groups and I was wondering if it could be generalised. In this case, the order of both direct products is the same, but the gcd and lcm of each of them are different.

My thought at this point is that they aren't isomorphic because the orders of their elements won't be the same if they have different gcd and lcm.

6. Re: Direct products and isomorphisms

Originally Posted by johng
I think Nehushtan's response is a little misleading. If m and n are not relatively prime, the direct product $\mathbb Z_m\times \mathbb Z_n$ is not cyclic.
Doh! Of course. I must have confused the cyclic groups $\mathbb Z_m,\mathbb Z_n$ with the subgroups $m\mathbb Z,n\mathbb Z$ of $\mathbb Z$.