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January 28th, 2017, 01:39 PM   #1
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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) ?
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February 2nd, 2017, 05:58 AM   #2
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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)
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February 2nd, 2017, 08:28 PM   #3
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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 $<a>\times <(5\,6)>$ has order 8, this subgroup is the centralizer.
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