it's about a regular n-gon which can be a pentagon for example

Fix an integer n3, let denote a regular convex n-gon, let V1,,Vn denote the vertices of , and let be a symmetry of . (a) If V is a vertex of , prove that (V) must also be a vertex of . Deduce that the n distinct vertices of can be expressed as (V1)(Vn). purpleHint: suppose, for the sake of contradiction, that (V) is not a vertex of . By Lemma 2.2.6, we can choose a vertex V for which d(V,V) is the maximal distance between any two vertices of . Use Lenma 2.2.8 to arrive at a contradiction. Proof: (b) If PR2 and (P) is a vertex of , use the result of part (a) to prove that P must also be a vertex of . Proof: (c) Suppose P,QR2. Prove that P and Q are adjacent vertices of if and only if (P) and (Q) are adjacent vertices of purpleHint: use the result of part (a), along with Lemma 2.2.6. Proof: (d) Suppose Vi and Vj are adjacent vertices of , and let be a symmetry of . Prove that = if and only if (Vi)=(Vi) and (Vj)=(Vj) purpleHint: let Vk be the other vertex adjacent to Vj. Use the result of parts (a) and (c) to prove that (Vk)=(Vk), and then apply the Corollary to Lemma 2.2.4. Proof: Fix an integer n3, let denote a regular convex n-gon, let V1,,Vn denote the vertices of , and let be a symmetry of . (a) If V is a vertex of , prove that (V) must also be a vertex of . Deduce that the n distinct vertices of can be expressed as (V1)(Vn). purpleHint: suppose, for the sake of contradiction, that (V) is not a vertex of . By Lemma 2.2.6, we can choose a vertex V for which d(V,V) is the maximal distance between any two vertices of . Use Lenma 2.2.8 to arrive at a contradiction. Proof: (b) If PR2 and (P) is a vertex of , use the result of part (a) to prove that P must also be a vertex of . Proof: (c) Suppose P,QR2. Prove that P and Q are adjacent vertices of if and only if (P) and (Q) are adjacent vertices of purpleHint: use the result of part (a), along with Lemma 2.2.6. Proof: (d) Suppose Vi and Vj are adjacent vertices of , and let be a symmetry of . Prove that = if and only if (Vi)=(Vi) and (Vj)=(Vj) purpleHint: let Vk be the other vertex adjacent to Vj. Use the result of parts (a) and (c) to prove that (Vk)=(Vk), and then apply the Corollary to Lemma 2.2.4. Proof