Let $ G $ be a group and $ a \in G $ an order element $ mn $, where $ m $ and $ n $ are relatively prime positive integers. Prove that there are $ x, y \in G $ such $ \vert x \vert = m $ and $ \vert y \vert = n $ and $ a = xy $.
Let $ G $ be a group and $ a \in G $ an order element $ mn $, where $ m $ and $ n $ are relatively prime positive integers
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abstract-algebra
group-theory
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0The notation to solve this may be a little easier if you think in terms of the cyclic group generated by $a$ being abelian (so you can use additive notation without loss of generality). – 2017-02-07
1 Answers
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Since $m$ and $n$ are relatively prime, we can write $1=mr+ns$ for some $r,s\in\mathbb{Z}$. Now, take $x=a^{ns}$ and $y=a^{mr}$ and show that this works...