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September 14th, 2017, 12:55 PM  #1 
Member Joined: Jan 2015 From: usa Posts: 67 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: 33 Thanks: 4 
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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