Let s be a permutation from Sn, for some n. Consider standard (unique) representation of s as a product s1s2...sk of independent cycles and let dec(s) be the decrement of s. Let (ij) be arbitrary transposition in S.
Verify that s(ij) is even, if s is odd, and that s(ij) is odd when s is even.
Consider the following cases:
1) both i,j are in the same cycle s1 and they are next to each other, i.e. s(i)=j.
2) both i,j are in the same cycle s1 and they are not next to each other.
3) i is in s1 and j is in s2.
4) i is in s1, and j is not in any cycle, i.e. s(j)=j.
5) both i,j are not moved by s, i.e. s(i)=i and s(j)=j.
Try to argue as general as possible, but you can start by checking particular cases to observe the pattern.
Remember that the independent cycles commute, so you can always assume that s1 is a first cycle in the composition (first on the right in the product).
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If there is any confusion with the wording please let me know, this is how it was worded when it was discussed in lecture.
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What I know:
I know that the decrement is equal to the number of moved elements minus the number of independent cycles. I also know that if σ∈Sn and dec(σ) is even then σ(ij) or ij(σ) is odd and if σ∈Sn and dec(σ) is odd then σ(ij) or ij(σ) is even (ij in these cases being an arbitrary transposition).
For part 2 of the cases, I know there is likely two subcases; one in which i,j are next to each other and another in which i,j are NOT next to each other. I know for the third case we wouldn't need to verify it both ways because of symmetry.
My issue here is really arguing for the general cases, as in I'm struggling with using particular cases to observe a pattern and then to establish general cases for the 5 listed cases.