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?
In my opinion (I guess I'll try to help you with the shape example if the following doesn't work) the most informative examples of group actions are when the groups act on themselves.
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