 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-i/7/a/iv/solution

For distinct

i, j \neq = 1

, direct conjugation gives

(ij) = (1 i) (1 j) (1 i) .

Thus the given star transpositions generate every transposition. Every cycle can in turn be expressed as

(a_{1} a_{2} \dots a_{ℓ}) = (a_{1} a_{ℓ}) (a_{1} a_{ℓ - 1}) \dots (a_{1} a_{2}) .

⟨(12), (13), \dots, (1 n)⟩ = S_{n} .

For

n = 1

S_{1}

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