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I tried to work out which cyclic subgroups an Abelian group $G$ contains if it contains an element of order $|a|$ and an element of order $|b|$.

I think the answer is that $G$ contains a cyclic subgroup for every divisor of $\text{lcm}(|a|,|b|)$. My resoning is that $G$ has to contain all the cyclic subgroups for divisors of $|a|$ and $|b|$. In addition to that it can contain subgroups generated by elements of the form $a^n b^m$. But all of these cases correspond to divisors of $\text{lcm}(|a|,|b|)$.

Please can you tell me if this is correct?

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You have just shown that some cyclic subgroups will be of order dividing $lcm(|a|,|b|)$. What you need to show is something stronger :

there exists an element of order $lcm(|a|,|b|)$

which is actually not so easy (but it is true).

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