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January 30th, 2018, 06:13 AM   #1
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Disjoint cycle decomposition in $A_n$

Please help me to prove the following result:


Show that if the disjoint cycle decomposition of $\sigma\in A_n$ includes a cycle of even lenght or two cycles of the same odd lenght then $C_{S_n}(\sigma)\not \subset A_n$

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Last edited by mona123; January 30th, 2018 at 06:32 AM.
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January 30th, 2018, 09:24 AM   #2
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I'm not really sure what $C_{S_n}(\sigma)$ is. Can you explain the notation?
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January 30th, 2018, 10:30 AM   #3
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the question that i am trying to answer now is the following: ($C_{S_n}$ is the centralizer)


Show that if the disjoint cycle decomposition of $\sigma\in A_n$ consists of cycles of odd lenght with no two lenghts the same then $C_{S_n}(\sigma) \subset A_n$

Can you please help me? thanks in advance
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January 30th, 2018, 10:38 AM   #4
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A conjugation of a disjoint cycle decomposition is very very easy: assume that $(a_1...a_n)...(b_1...b_m)$ is a disjoint cycle decomposition, then
$$\alpha \circ (a_1...a_n)...(b_1...b_m) \circ \alpha^{-1} = (\alpha(a_1) .... \alpha(a_n)) .... (\alpha(b_1) ... \alpha(b_m))$$

Now relate the centralizer to conjugations and it becomes very easy.
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January 30th, 2018, 10:43 AM   #5
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i don't see clearly how to use what you wrote, can you please explain more ?
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January 31st, 2018, 05:34 PM   #6
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Here's an outline of a proof. I think you should make certain that you understand the statements.

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