# Isomorphism between groups, direct product, lcm, and gcd

1. Jan 25, 2009

### Daveyboy

1. The problem statement, all variables and given/known data

Let a,b, be positive integers, and let d=gcd(a,b) and m=lcm(a,b). Show ZaXZb isomorphic to ZdXZm

2. Relevant equations

m=lcm(a,b) implies a|m, b|m and if a,b|c then m|c.
d=gcd(a,b) implies d|a, d|b and if c|a and c|b then d|c

let n be an integer with prime decomposition n=p1a1...pkak
then Zn=Zp1a1X...XZp1ak
3. The attempt at a solution

It is clear that d|m

I considered the prime decomposition of a and b as sets, denote as A and B respectively, and consider the intersection, call this set P. intersection(A,B) = P

Call the product of the elements in this set P'. I claim P' = gcd(a,b), this should be clear.

Then I considered the lcm as the product of the elements in A-P and B-P call these sets A' and B' respectively.

Now we can write the d = lcm(a,b) =A'*B'*P' and this should be clear.

So... now I want to use the isomorphism as described as in the section above, but I don't know what to do. I think I should use ZaXZb is isomorphic to something like
(*)
Zp1k1X...XZpikiXZq1l1X...XZpnln.

where p are the primes of a and q are the primes of b.

Great and now I want to show that that ZdXZm is isomorphic to the same thing, but I'm starting to think that it is not.

I need to make an argument stronger that one of just cardinality, but I don't know what. I'm feel that I need to justify stuff in (*) but I'm not even sure if it is true!

It took me a long time to think of this and I really need to get it done... help would be greatly appreciated.