Reference for theorem on quotient of generalized quaternion group by its centre being isomorphic to dihedral group

72 Views Asked by At

I am looking for a book/paper reference for the following theorem;

Suppose $ n \geq 3 $. Let $ Q_{4n} $ be the generalized quaternion group of order $ 4n $. Then $ | Z(Q_{4n} ) | = 2 $ and $ Q_{4n} / Z(Q_{4n} ) \cong D_{2n} $, where $ D_{2n} $ is the dihedral group of order $ 2n $.

I have looked about but can't find one myself. There is a decent proof here https://planetmath.org/generalizedquaterniongroup

However I would prefer not to link to a website since they can disappear quite easily and I want to avoid writing out the full proof myself.

Edit: Would also prefer to find a reference that is not left as an exercise.