1
$\begingroup$

This question already has an answer here:

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.

$\endgroup$

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ I would suggest starting by looking at D8 and finding by hand all elements that commute with everything. $\endgroup$ – ufabao Mar 26 '16 at 20:32
-1
$\begingroup$

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 $rs=(rs)^{-1}=s^{-1}r^{-1}=s^2r \neq sr$.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.