Suppose $G$ is an abelian, and $g,h∈G$. Suppose $g$ and $h$ both have finite order $m$ and $n$. Is there any example that the order of $gh$ is not the least common multiple of $m$ and $n$?
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11what if $g=h^{-1}$? – Angina Seng Oct 07 '18 at 18:54
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1Related https://math.stackexchange.com/questions/67180/order-of-product-of-two-elements-in-a-group – TrostAft Oct 07 '18 at 18:58
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2$1\bmod 8$ and $3\bmod 8$ have order $8$ each, but their sum $4\bmod 8$ has order $2$. $1\mod 6$ has order $6$, $2\mod 6$ has order $3$, their sum has order $2$. -- It seems you did not try many simple cases (here, small cyclic groups) – Hagen von Eitzen Oct 07 '18 at 18:59
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@HagenvonEitzen Thank you, it is extremely helpful – JJW22 Oct 07 '18 at 19:04