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 September 14th, 2017, 12:55 PM #1 Senior Member   Joined: Jan 2015 From: usa Posts: 101 Thanks: 0 Abelinization of a group I want identify the abelianization $Q_{4n}/Q'_{4n}$ with a familiar group where $Q_{4n}$ is the dicyclic group. Thanks.
 September 16th, 2017, 11:14 AM #2 Member   Joined: May 2017 From: Russia Posts: 34 Thanks: 5 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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