Paper 3 , Section II, D

Groups | Part IA, 2009

Let S(X)S(X) denote the group of permutations of a finite set XX. Show that every permutation σS(X)\sigma \in S(X) can be written as a product of disjoint cycles. Explain briefly why two permutations in S(X)S(X) are conjugate if and only if, when they are written as the product of disjoint cycles, they have the same number of cycles of length nn for each possible value of nn.

Let (σ)\ell(\sigma) denote the number of disjoint cycles, including 1-cycles, required when σ\sigma is written as a product of disjoint cycles. Let τ\tau be a transposition in S(X)S(X) and σ\sigma any permutation in S(X)S(X). Prove that (τσ)=(σ)±1\ell(\tau \sigma)=\ell(\sigma) \pm 1.

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