Paper 3, Section I, D

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

Describe a condition on the centraliser (in $S_{n}$ ) of a permutation $\sigma \in A_{n}$ that ensures the conjugacy class of $\sigma$ in $A_{n}$ is the same as the conjugacy class of $\sigma$ in $S_{n}$. Justify your answer.

How many distinct conjugacy classes are there in $A_{5}$ ?

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