My Math Forum Disjoint cycle decomposition in $A_n$

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 January 30th, 2018, 05:13 AM #1 Senior Member   Joined: Jan 2015 From: usa Posts: 103 Thanks: 0 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 $C_{S_n}(\sigma)\not \subset A_n$ Thanks Last edited by mona123; January 30th, 2018 at 05:32 AM.
 January 30th, 2018, 08:24 AM #2 Senior Member   Joined: Oct 2009 Posts: 555 Thanks: 179 I'm not really sure what $C_{S_n}(\sigma)$ is. Can you explain the notation?
 January 30th, 2018, 09:30 AM #3 Senior Member   Joined: Jan 2015 From: usa Posts: 103 Thanks: 0 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
 January 30th, 2018, 09:38 AM #4 Senior Member   Joined: Oct 2009 Posts: 555 Thanks: 179 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.
 January 30th, 2018, 09:43 AM #5 Senior Member   Joined: Jan 2015 From: usa Posts: 103 Thanks: 0 i don't see clearly how to use what you wrote, can you please explain more ?
 January 31st, 2018, 04:34 PM #6 Member   Joined: Jan 2016 From: Athens, OH Posts: 89 Thanks: 47 Here's an outline of a proof. I think you should make certain that you understand the statements.

 Tags $an$, cycle, decomposition, disjoint

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