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Suppose that $a, b, c$ and $d$ are positive integers such that $b$ is an integer multiple of $a$, and $d$ is an integer multiple of $c$. How can we prove that

if the direct sums $ \mathbb Z_a\oplus \mathbb Z_b $ and $\mathbb Z_c\oplus \mathbb Z_d $ are isomorphic then $a=c$ and $b=d$.

What I have done is:

If $b$ is multiple of $a$, then there exists an integer $m$ such that $ b=a\cdot m $. Similarly, if $d$ is an integer multiple of $c$, there exists an integer $n$ such that $ d=c\cdot n $

If $ \mathbb Z_a\oplus \mathbb Z_b $ and $\mathbb Z_c\oplus \mathbb Z_d $ are isomorphic, then $ a\cdot b=c\cdot d $

Then we get $ a^2\cdot m= c^2\cdot n $ . But it seems like we can not get anything from this to reach the answer.

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    Consider the largest order of an element in either group.2012-02-25
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    $\mathbb{Z}_a$ is the cyclic group of $a$ elements? In your spot, the next thing I'd think to try is to use your equations to make a counter-example. (and if I can't, figure out why I fail) It might help to work one prime at a time -- i.e. $a,b,c,d$ all be powers of the same prime. (if you can do this, you can use Chinese Remainder Theorem or somesuch for the full answer)2012-02-25
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    @Henning Makholm: $\phi(b)=\phi(d)$ right?2012-02-25
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    @ArtiomFiodorov: _Additive_ order.2012-02-25
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    @HenningMakholm thanks a lot2012-02-27

1 Answers 1

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HINTS:

  1. If $a\mid b$, what is the largest order of any element of $\mathbb{Z}_a\oplus\mathbb{Z}_b$? Is this number an isomorphism invariant?

  2. What is the order of $\mathbb{Z}_a\oplus\mathbb{Z}_b$?

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    if b=am then the largest order of any element of Za⊕Zb is am. Similarly if d=cn then the largest order of any element of Zc⊕Zd is cn. Since they are isomorphic then am=cn. If Za⊕Zb and Zc⊕Zd are isomorphic, then a⋅b=c⋅d since the isomorphic groups have the same order. So, a^2.m=c^2.n and am=cn we can see that a=c.2012-02-27
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    thanks a lot for the help.2012-02-27