Hint request: Find all elements of the group of symmetries of the octagon which commute with all other members of the group.

I'd like to request a hint

So far, I've enumerated all 16 symmetries. I know that $D_8$ has order $16$, so I've got them all. I know that I could in theory just check them all, which would be tedious but doable. I'm sure there is a trick to this though. Unfortunately, I do not have any other ideas. I know that in general, a rotation and a reflection do not commute, but I'm not sure if I can extend that to every rotation and reflection.

Obviously I have that the identity commutes.

If someone could give a hint that would be much appreciated.

Note that any diehedral group $D_n$, can be described as being generated by a generator $a$ of order $n$ and a generator $b$ of order $2$, bound by the relation $b^{-1}ab=a^{-1}$. Let $a^i$ be an element of the center of $D_n$ (i.e. an element that commutes with all other elements). It certainly already commutes with all powers of $a$ so we have to inspect when it commutes with $b$. This happens when $b^{-1}a^ib = a^{i}$ or $a^{-i} = a^{i}$, or $a^{2i}=1$. So $n$ has to be even and the commuting rotation is given by $a^{\frac{n}{2}}$, which correponds to a $180°$ turn.
• Every reflection has the form $s = a^ib$. So the condition for s to commute with every rotation $a^j$ is $s^{-1}a^js = a^j$, so we must have $b^{-1}a^{-i}a^ja^ib = a^j$, or $b^{-1}a^jb = a^j$ or $a^{-j} = a ^j$. This is clearly not true for all possible $j$. PS: I would much appreciate it if you mark my answer as accepted''. – Marc Bogaerts Oct 6 '16 at 17:54
• Ok. After playing around with a cut out octagon for a while, I'm pretty sure that rotating $180^{\circ}$ works. I still really have no clue how to prove that in any way other than checking it for every other element. Not to mention that I still can't even convince myself! I don't think that there are any reflections that will work, but again I'm not able to prove this. – Alex Oct 5 '16 at 22:16