# If $\mathbb{Z}_a\oplus\mathbb{Z}_b\cong \mathbb{Z}_c\oplus\mathbb{Z}_d$, $a|b$, and $c|d$, then $a=c$ and $b=d$.

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. – Henning Makholm Feb 25 '12 at 23:11
$\mathbb{Z}_a$ 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) – Hurkyl Feb 25 '12 at 23:14
@Henning Makholm: $\phi(b)=\phi(d)$ right? – Tom Artiom Fiodorov Feb 25 '12 at 23:21
@ArtiomFiodorov: Additive order. – Henning Makholm Feb 25 '12 at 23:25
@HenningMakholm thanks a lot – the code Feb 27 '12 at 22:02

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 $\mathbb{Z}_a\oplus\mathbb{Z}_b$?