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Isomorphic groups

Groups, Rings, Domains, Modules, etc, Galois theory
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Riemann
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Isomorphic groups

#1

Post by Riemann » Sun Jul 16, 2017 4:57 pm

Let $n >2$. Define the group

$$\mathcal{Q}_{2^n} = \langle x, y \mid x^2=y^{2^{n-2}} , y^{2^{n-1}} = 1, x^{-1} yx =y^{-1} \rangle$$

Show that $\mathcal{Q}_{2^n} / \mathcal{Z} \left ( \mathcal{Q}_{2^n} \right ) \simeq \mathcal{D}_{2^{n-1}}$ where $\mathcal{D}$ is the dihedral group.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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