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I need to prove/disprove whether two groups are isomorphic or not :

The two groups are : $Z_{18} \times Z_{15} \times Z_{12} $ and $Z_{90} \times Z_{36}$

Now using the property that $Z_{m} \times Z_{n}$ is isomorphic to $Z_{mn}$ iff $m $ and $n$ are co prime the first group can be transformed to

$Z_{90} \times Z_{3} \times Z_{12}$

Now using the Fundamental theorem of Finite Abelian Groups : states that" a finite Abelian group is isomorphic to a direct product of cyclic groups of prime-power order, where the decomposition is unique up to the order in which the factors are written."

Now Clearly by any means I cannot Further decompose/combine different groups to give $Z_{3} \times Z_{12}$ as $Z_{36}$

So by using above theorem two groups must be non -isomorphic.

Is my reasoning and answer correct ?

Shaun
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    I don't see where you show that we arrive at $\Bbb Z/90 \times \Bbb Z/3\times \Bbb Z/12$, but the idea is correct (and often used, see for example here etc.). You should say, that you use it for $m=18$ and $n=5$. – Dietrich Burde Oct 19 '19 at 18:51
  • @DietrichBurde: well, $Z_{18} \times Z_{15} \times Z_{12}$ = $Z_{18} \times Z_{5} \times Z_{3} \times Z_{12} $ = $Z_{90} \times Z_{3} \times Z_{12} $
    Should I update this on my question ?     –  Oct 19 '19 at 19:17

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