How can i show a group action where G is the group of symmetries of a regular octagon and X is the set of all possible colourings of a regular octagon where each edge is coloured one of two colours??
And could someone please provide more examples like this one, may be using a different shape e.g. a square or a triangle?
For example, remember Lagrange's theorem? It can be said more "eloquently" in the language of group actions.
So, let be a group and . Then we can define a group action on by h,g)\mapsto hg" alt="\alpha:H\times G\to Gh,g)\mapsto hg" />. This is a group action since and . Now, think of the orbits of the relation as . It is easily shown that that . Thus, since and it follows that which implies that