3.II.6E3 . \mathrm{II} . 6 \mathrm{E} \quad

Groups | Part IA, 2008

Prove that two elements of SnS_{n} are conjugate if and only if they have the same cycle type.

Describe (without proof) a necessary and sufficient condition for a permutation σAn\sigma \in A_{n} to have the same conjugacy class in AnA_{n} as it has in SnS_{n}.

For which σSn\sigma \in S_{n} is σ\sigma conjugate (in SnS_{n} ) to σ2?\sigma^{2} ?

For every σA5\sigma \in A_{5}, show that σ\sigma is conjugate to σ1\sigma^{-1} (in A5)\left.A_{5}\right). Exhibit a positive integer nn and a σAn\sigma \in A_{n} such that σ\sigma is not conjugate to σ1\sigma^{-1} (in AnA_{n} ).

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