I have to find the order of the elements of the dihedral group $\displaystyle D_4 = \{1,y,y^2,y^3,x,xy,xy^2,xy^3\} $ but I don't really understand what this group is even though I've read my notes 10 times. It must simply be the way it is explain that doesn't fit my mind. So how can we find this and why is it like this?