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5.2: Dihedral Groups
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/05%3A_Permutation_Groups/5.02%3A_Dihedral_Groups
Let $s = s_1\text{.}$ Then $s^2 = 1$ and $r^n = 1\text{.}$ Since any rigid motion $t$ of the $n$ -gon replaces the first vertex by the vertex $k\text{,}$ the second vertex must be replaced ...Let $s = s_1\text{.}$ Then $s^2 = 1$ and $r^n = 1\text{.}$ Since any rigid motion $t$ of the $n$ -gon replaces the first vertex by the vertex $k\text{,}$ the second vertex must be replaced by either $k+1$ or by $k-1\text{.}$ If the second vertex is replaced by $k+1\text{,}$ then $t = r^k\text{.}$ If the second vertex is replaced by $k-1\text{,}$ then $t = r^k s\text{.}$ 2 Hence, $r$ and $s$ generate $D_n\text{.}$ That is, $D_n$ consists of all finite products of \…

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