# find the center of D8, D10 and Dn [duplicate]

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.

## marked as duplicate by Antonios-Alexandros Robotis, user228113, zz20s, Adam Hughes, Dietrich BurdeMar 26 '16 at 20:52

Hint: There is a standard presentation of $Dn$ 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 $<r, s: r^2=s^3=1, (rs)^2=1 >$. Now $s$ and $r$ do not commute because $rs=(rs)^{-1}=s^{-1}r^{-1}=s^2r \neq sr$.