In general, .
By the Chinese Remainder Theorem,
if gcd(p,q) > 1, then ℤ_{p} x ℤ_{q} ≇ ℤ_{pq}
Does this imply that if ℤ_{a} x ℤ_{b }≇ ℤ_{c} and ℤ_{u}x ℤ_{v }≇ ℤ_{c}, then ℤ_{a} x ℤ_{b }≇ ℤ_{u}x ℤ_{v} ?
Or is there something I'm missing?
Thanks!
Hi,
First, I think Nehushtan's response is a little misleading. If m and n are not relatively prime, the direct product is not cyclic.
Next, I really don't know what you're asking. The relation "not isomorphic" is never transitive.
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.