Let G be the group of symmetries (including flips) of the regular heptagon (7-gon) shown at...

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Let G be the group of symmetries (including flips) of the regular heptagon (7-gon) shown at right. As usual, we regard the elements of G as permutations of the set of vertex labels: thus, G < S7. (a) Let o denote the rotation of the 7-gon that takes the vertex 1 to the vertex 2. Write down the cyclic subgroup R: (a) as a set of elements of S in cycle notation. (b) What are the orders of each of the elements of R? (c) Write down any symmetry EGR in cycle notation. (d) Find the left coset R for your chosen p from (c). What are the orders of each of the elements of R? 4 (e) According to Lagrange's Theorem, what are the possible values of H, where H < G? (f) Let H be any subgroup of G other than G itself. Using (e), explain why H is cyclic. (g) Determine the number of subgroups of G (including G). Explain your answer. Let G be the group of symmetries (including flips) of the regular heptagon (7-gon) shown at right. As usual, we regard the elements of G as permutations of the set of vertex labels: thus, G < S7. (a) Let o denote the rotation of the 7-gon that takes the vertex 1 to the vertex 2. Write down the cyclic subgroup R: (a) as a set of elements of S in cycle notation. (b) What are the orders of each of the elements of R? (c) Write down any symmetry EGR in cycle notation. (d) Find the left coset R for your chosen p from (c). What are the orders of each of the elements of R? 4 (e) According to Lagrange's Theorem, what are the possible values of H, where H < G? (f) Let H be any subgroup of G other than G itself. Using (e), explain why H is cyclic. (g) Determine the number of subgroups of G (including G). Explain your answer.