How would you show the group action where G is the symmetries of a regular octagon and X is a set of all possible coulourings of a regular octagon where each edge is coloured one of two colours?
Hi, I have tried some of key steps of your problem. You need to fill the remaining steps. Assume the same color can be used on any number of edges.
Burnside's lemma
$\displaystyle r \cdot |G| = \sum_{g \in G}|X_g|$.
The number of distinguishable regular octagons correspond to the number of orbits under G in a G-set X.
To apply symmetry to a regular octagon, we use $\displaystyle G = D_8=\{1, r, r^2, r^3, ..., r^7, s, sr, sr^2, sr^3, ..., sr^7\}$.
Now we need to compute $\displaystyle |X_g|$ for each sixteen elements g in $\displaystyle D_8$.
$\displaystyle |X_1| = 2^8$. (Every regular octagon is left fixed by the identity element)
$\displaystyle |X_r| = |X_{r^7}| = 2$. (To be invariant under r, r^7, all edges should be the same color. There are two colors, so two choices)
$\displaystyle |X_{r^4}| = 6$. (To be invariant under r^4, all edges should be same colors (oooooooo, xxxxxxxx), or different adjacent colors (oxoxoxox, xoxoxoxo), and (ooxxooxx, xxooxxoo). Here , (oxoxoxox) denotes edge 1 (between vertex 1 and 2) is colored o, edge 2 is colored x (between vertex 2 and 3), and so on.
$\displaystyle |X_s| = 4$. Diagonal flip : (Possible choices: oooooooo, xxxxxxxx, oxoxoxox, xoxoxoxo}
.....
After you find all |X_g| values, you need to sum up those |X_g| values
$\displaystyle \sum_{g \in G}|X_g| = \{2^8 + 2 + .. + 6 + ..+ 4 + ...\}$
Finally, the number of orbits N (distinguishable regular octagons) can be acquired.
$\displaystyle N = \frac{\sum_{g \in G}|X_g|}{|D_{8}| = 16}$.