# Group of symmetries on a regular polygon

by SiddharthM
Tags: polygon, regular, symmetries
 P: 291 Here is how to think of it. 1)Draw a regular polygon with vertices 1,2,...,n in order. 2)The question is how many ways can the polygon be replaced by rigid motions, i.e. with vertices still in order. Now for any one of the n locatations there are n choices. Having chosen that there is only one chose for the next one in succession, i.e. either left or right. Having chosen that the polygon is completely determined. Thus there are 2n elements in this group. 3)Let a be the positive rotation by 2pi/n i.e. the cycle (1,2,...,n) and let b be the reflection through the middle and vertex 1, i.e. (2,n)(3,n-1)..... 4)So a^n = 1 and b^2 = 1. 5)Consider S = {a^k,ab^k} for k=1,2,...,n 6)There are 2n elements in that set all of which are distinct. So S represents the group D_n. 7)Now it remains to show ba=a^{n-1}b which is not so hard to show. Hence the group presentation for the dihedral group is: $$\left< a,b| a^n = 1, \ b^2 = 1, \ ba=a^{n-1}b \right>$$