The center of a group $G$ is $$Z(G) = \{g \in G\ :\ \forall x\in G,\ gx = xg\}$$ Find the center of D8. What about the center of D10? What is the center of Dn?
I am unsure where to start for this problem.
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I would suggest starting by looking at D8 and finding by hand all elements that commute with everything. – ufabao Mar 26 '16 at 20:32
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For odd $n$ it's centerless; for even it's $\Bbb Z_2,$ generated by the rotation through the angle $\pi.$ use the relators to write a proof. – suckling pig Aug 24 '24 at 07:11
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Hint: There is a standard presentation of $D_n$ as a group with two generators. Using this presentation, try to see what conditions you need on an arbitrary element for it to commute with all other elements of the group.
For instance, $S_3$ can be written as $\langle r, s: r^2=s^3=1, (rs)^2=1 \rangle$. Now $s$ and $r$ do not commute because $rs=(rs)^{-1}=s^{-1}r^{-1}=s^2r \neq sr$.
Alexander
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