Prove additive group Z6 × Z25 × Z49 is cyclic. Just need to know exact steps of how I should think about this question, what properties if the groups should I be focusing on.
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2Does this answer your question? proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where $l=lcm(m,n$) and $d=gcd(m,n)$ – Arnaud D. Apr 20 '20 at 21:44
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Hint: Chinese remainder theorem. – Robert Israel Apr 20 '20 at 20:07
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The order of element $(a_1,a_2,\dots,a_n)$ in group $A_1 \times A_2 \dots A_n$ is the lcm of the orders of the $a_i$.
If these orders are coprime then the order of the element is the product of the orders.
In our example we can find an element of order $6\times 25\times 49$ so it is cyclic.
In general one has a product of cyclic groups is cyclic if and only if the orders are coprime.
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