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Mathematics 1214: Introduction to Group Theory Tutorial exercise sheet 8 1. For each element α of the dihedral group D3 , compute the order o(α) and find hαi. 2. Let G be an abelian group. Prove that if a, b ∈ G and n ∈ Z, then (ab)n = an bn = bn an . [Hint: you might want to consider the three cases n = 0, n > 0 and n < 0 separately.] 3. Prove that if G is a group and a, b ∈ G and n ∈ Z, then (ab)n a = a(ba)n . [Hint: you have to prove this for all groups, not just abelian groups. So make sure you don’t assume that G is abelian.] 4. Consider the abelian group G = ((0, ∞), ·) of the positive real numbers under multiplication. Prove that G is not a cyclic group. 5. Let n ≥ 3 and consider the dihedral group Dn = {ι, ρ, ρ2 , . . . , ρn−1 , r0 , r1 , r2 , . . . , rn−1 } where ι is the identity mapping on P , ρ is rotation by 2π/n counterclockwise about the origin, and for 0 ≤ j < n, the mapping rj is reflection in the straight line through the origin making an angle of π/n with the positive x-axis. (a) Find: • a cyclic subgroup of Dn of order n; • n cyclic subgroups of Dn of order 2; • a cyclic subgroup of Dn of order 1. (b) Explain why Dn is not a cyclic group. (c) Disprove the following statement: for every n ≥ 3, the only non-cyclic subgroup of Dn is Dn itself.