 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/8/c/solution

Each reflection has order two, giving exactly the three order-two subgroups

⟨ s ⟩ = {e, s}, ⟨ rs ⟩ = {e, rs}, ⟨ r^{2} s ⟩ = {e, r^{2} s} .

r

sends

r^{j} s

r^{j + 2} s

, cycling the three reflections. None of these subgroups is therefore normal.

The two nonidentity rotations generate the same order-three subgroup:

R = ⟨ r ⟩ = {e, r, r^{2}} ⊴ D_{3} .

It is invariant under conjugation by

r

, and

srs = r^{- 1}

shows invariance under

s

as well; since

r, s

generate the group, this proves normality. The complete partition into conjugacy classes is

{e}, {r, r^{2}}, {s, rs, r^{2} s} .

The rotations are mutually conjugate by

s

, and the reflections by

r

; no rotation is conjugate to a reflection because conjugation preserves the order of a group element.

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