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January 28th, 2017, 12:39 PM  #1 
Newbie Joined: Jan 2017 From: Nowhere Posts: 5 Thanks: 1  Find all permutations that commute with given permutation
Can someone help me find all the permutations in S6 that comute with (1 3 4 2) ?

February 2nd, 2017, 04:58 AM  #2 
Senior Member Joined: Feb 2012 Posts: 144 Thanks: 16 
here are some hints: 1) the permutations which commute with a given permutation form a subgroup of Sn (prove it!) 2) obviously, any permutation x commutes with itself, hence the subgroup generated by x is included in the subgroup you're looking for. maybe you could begin by computing the subgroup generated by your permutation (the one you give shows only 4 numbers, so it is not a permutation in S6) 
February 2nd, 2017, 07:28 PM  #3 
Member Joined: Jan 2016 From: Athens, OH Posts: 93 Thanks: 48 
There are $6\cdot5\cdot3=90$ 4cycles in $S_6$; since any 4cycle is conjugate to your 4cycle $a=(1\,3\,4\,2)$ the index of the centralizer of $a$ in $S_6 $ is 90. Thus the order of the centralizer is 8. Since $<a>\times <(5\,6)>$ has order 8, this subgroup is the centralizer.


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