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 January 28th, 2017, 01:39 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, 05:58 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, 08:28 PM #3 Member   Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 There are $6\cdot5\cdot3=90$ 4-cycles in $S_6$; since any 4-cycle is conjugate to your 4-cycle $a=(1\,3\,4\,2)$ the index of the centralizer of $a$ in $S_6$ is 90. Thus the order of the centralizer is 8. Since $\times <(5\,6)>$ has order 8, this subgroup is the centralizer.

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