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

Question

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.

Answer 1

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.

Answer 2

Draw some pictures and try to understand why exactly the rotations and reflections do not commute (do it for a general regular polygon, boring otherwise). This will help you to understand what kind of rotation you should pick up as a candidate for the central (commuting with everyone) element.