 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-i/5/b/ii/solution

S_{3}

, composing permutations from right to left. Set

x = (12)

and

y = (23)

x^{3} = x

and

y^{3} = y

, whereas

x y = (123)

has order of a group element three. Consequently

T (x y) = (123)^{3} = e, T (x) T (y) = (12) (23) = (123) \neq = e .

Thus the cubing map on

S_{3}

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