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epr2008

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## Homework Statement

If G is a group with operation * and [tex]\alpha,\beta\in G[/tex], then [tex]\beta\ast\alpha\ast\beta^{-1}[/tex] is called a conjugate of G. Compute the number of conjugates of each 3-cycle in S[tex]_{n}[/tex] (n[tex]\geq[/tex]3).

## Homework Equations

## The Attempt at a Solution

For any group [tex]S_{n}[/tex] there must be [tex]_{n}P_{m}[/tex] m-cycles. Each m-cycle has m permutations. Then, the number of m-cycles in [tex]S_{n}[/tex] is [tex]\frac{_{n}P_{m}}{m}[/tex]. Since an m-cycle exchanges m objects, it's inverse must also exchange m-objects. So, the inverse must also be an m-cycle.

I have an idea of how to go on from here, but I am not sure about it since some m-cycles are their own inverse...

If m is odd then there are an even number of m-cycles in [tex]S_{n}[/tex]. Then it must be that for odd m, any m-cycle [tex]\beta\neq\beta^{-1}[/tex]. Therefore, for odd m, there are [tex]\frac{_{n}P_{m}}{2m}[/tex] conjugates of order m in [tex]S_{n}[/tex]. In the case of the 3-cycle, there would be [tex]\frac{_{n}P_{3}}{2\cdot3}=\frac{n(n-1)(n-2)}{6}[/tex] conjugates.

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