Paper 3, Section I, E

What is a cycle in the symmetric group $S_{n}$ ? Show that a cycle of length $p$ and a cycle of length $q$ in $S_{n}$ are conjugate if and only if $p=q$.

Suppose that $p$ is odd. Show that any two $p$-cycles in $A_{p+2}$ are conjugate. Are any two 3 -cycles in $A_{4}$ conjugate? Justify your answer.

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