(adsbygoogle = window.adsbygoogle || []).push({}); 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_{a}XZ_{b}isomorphic to Z_{d}XZ_{m}

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=p_{1}^{a1}...p_{k}^{ak}

then Z_{n}=Z_{p1a1}X...XZ_{p1ak}

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 Z_{a}XZ_{b}is isomorphic to something like

(*)

Z_{p1k1}X...XZ_{piki}XZ_{q1l1}X...XZ_{pnln}.

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

Great and now I want to show that that Z_{d}XZ_{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.

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# Homework Help: Isomorphism between groups, direct product, lcm, and gcd

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