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January 30th, 2018, 06:13 AM  #1 
Member Joined: Jan 2015 From: usa Posts: 92 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 06:32 AM. 
January 30th, 2018, 09:24 AM  #2 
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 84 
I'm not really sure what $C_{S_n}(\sigma)$ is. Can you explain the notation?

January 30th, 2018, 10:30 AM  #3 
Member Joined: Jan 2015 From: usa Posts: 92 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, 10:38 AM  #4 
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 84 
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, 10:43 AM  #5 
Member Joined: Jan 2015 From: usa Posts: 92 Thanks: 0 
i don't see clearly how to use what you wrote, can you please explain more ?

January 31st, 2018, 05:34 PM  #6 
Member Joined: Jan 2016 From: Athens, OH Posts: 79 Thanks: 39 
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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