My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
January 30th, 2018, 05:13 AM   #1
Senior Member
 
Joined: Jan 2015
From: usa

Posts: 101
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.
mona123 is offline  
 
January 30th, 2018, 08:24 AM   #2
Senior Member
 
Joined: Oct 2009

Posts: 436
Thanks: 147

I'm not really sure what $C_{S_n}(\sigma)$ is. Can you explain the notation?
Micrm@ss is offline  
January 30th, 2018, 09:30 AM   #3
Senior Member
 
Joined: Jan 2015
From: usa

Posts: 101
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
mona123 is offline  
January 30th, 2018, 09:38 AM   #4
Senior Member
 
Joined: Oct 2009

Posts: 436
Thanks: 147

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.
Micrm@ss is offline  
January 30th, 2018, 09:43 AM   #5
Senior Member
 
Joined: Jan 2015
From: usa

Posts: 101
Thanks: 0

i don't see clearly how to use what you wrote, can you please explain more ?
mona123 is offline  
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.

johng40 is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
$an$, cycle, decomposition, disjoint



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
which one is more efficience otto cycle or carnot cycle or d r-soy Physics 0 June 2nd, 2013 02:14 AM
Product of disjoint cycles Solarmew Applied Math 1 April 12th, 2012 09:01 AM
pairwise almost disjoint. kleopatra Applied Math 0 June 26th, 2009 10:47 AM
union of disjoint cosets stf123 Abstract Algebra 3 October 9th, 2007 07:29 PM





Copyright © 2018 My Math Forum. All rights reserved.