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

View attachment 8446

As usual, we regard the elements of G as permutations of the set of vertex labels; thus, G ≤ S7.

(a) Let σ denote the rotation of the 7-gon that takes the vertex 1 to the vertex 2. Write down the cyclic subgroup R := ⟨σ⟩ as a set of elements of S7 in cycle notation.

(b) What are the orders of each of the elements of R?

does this mean R := ⟨( 1 2 3 4 5 6 7 )⟩?

and I am unsure how to answer part b)

thanks

View attachment 8446

As usual, we regard the elements of G as permutations of the set of vertex labels; thus, G ≤ S7.

(a) Let σ denote the rotation of the 7-gon that takes the vertex 1 to the vertex 2. Write down the cyclic subgroup R := ⟨σ⟩ as a set of elements of S7 in cycle notation.

(b) What are the orders of each of the elements of R?

does this mean R := ⟨( 1 2 3 4 5 6 7 )⟩?

and I am unsure how to answer part b)

thanks