Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
6  
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
2  
@ArtiomFiodorov: Additive order. –  Henning Makholm Feb 25 '12 at 23:25
    
@HenningMakholm thanks a lot –  the code Feb 27 '12 at 22:02
add comment

1 Answer

up vote 1 down vote accepted

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

share|improve this answer
    
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. –  the code Feb 27 '12 at 22:00
    
thanks a lot for the help. –  the code Feb 27 '12 at 22:01
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.