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September 14th, 2017, 12:55 PM   #1
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Abelinization of a group

I want identify the abelianization $Q_{4n}/Q'_{4n}$ with a familiar group where $Q_{4n}$ is the dicyclic group.

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September 16th, 2017, 11:14 AM   #2
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Can we regard $\displaystyle Q_{4n}$ as
$\displaystyle \langle a,\ x\ | \ a^{2n}=e, \ x^2=a^{n}, \ xax^{-1}=a^{-1} \rangle$ ?

https://groupprops.subwiki.org/wiki/Dicyclic_group

I think you could try to prove that any commutator of $\displaystyle Q_{4n}$ can be written as $\displaystyle a^{2k},\ k\in\mathbb{N}$.
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