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.