Isomorphism between groups, direct product, lcm, and gcd

[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 Z_aXZ_b isomorphic to Z_dXZ_m

Homework 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=p₁^a₁...p_k^a_k
then Z_n=Z_p1^a1X...XZ_p1^ak

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 Z_aXZ_b is isomorphic to something like
(*)
Z_p1^k1X...XZ_pi^kiXZ_q1^l1X...XZ_pn^ln.

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

Great and now I want to show that that Z_dXZ_m 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.

 

[mighty2000]

Hello!

First, let's review the definitions of gcd and lcm:

-gcd(a,b) = the greatest positive integer that divides both a and b.
-lcm(a,b) = the smallest positive integer that is divisible by both a and b.

Now, let's look at the sets A and B as you defined them. A = {p1, p2, ..., pk} and B = {q1, q2, ..., ql}. Then, the intersection A∩B = {p1, p2, ..., pk, q1, q2, ..., ql}, which we can write as P = {p1, p2, ..., pk, q1, q2, ..., ql}.

Next, we can write A' = A-P and B' = B-P, which means that A' = {p1, p2, ..., pk} and B' = {q1, q2, ..., ql}. Then, the product A'*B' = {p1, p2, ..., pk}*{q1, q2, ..., ql} = {p1q1, p1q2, ..., p1ql, p2q1, p2q2, ..., p2ql, ..., pkq1, pkq2, ..., pkql}.

Since P' = gcd(a,b), we know that P' divides both a and b. Therefore, A'*B'*P' = {p1q1, p1q2, ..., p1ql, p2q1, p2q2, ..., p2ql, ..., pkq1, pkq2, ..., pkql}*gcd(a,b) = {p1q1gcd(a,b), p1q2gcd(a,b), ..., p1qlgcd(a,b), p2q1gcd(a,b), p2q2gcd(a,b), ..., p2qlgcd(a,b), ..., pkq1gcd(a,b), pkq2gcd(a,b), ..., pkqlgcd(a,b)}.

Now, let's look at the sets (*) and ZdXZm. In (*) we have Zp1k1X...XZpikiXZq1l1X...XZpnln, which is the same as Zpk1X...XZpikiXZql1X...XZqln. This is the same as Zpk1X...XZ