- #1
Buri
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My text defines an elementary permutation as follows:
Give 1 <= i < k, let e_i be the element of S_k (where S_k is the set of all permutations of {1,...,k}) defined by setting e_i (j) = j for j not equal to i,i+1; and e_i (i) = i + 1 and e_i (i + 1) = i.
It goes on to say that if f is in S_k, then it can be written as a composite of elementary permutations. But I don't see how {k,2,3,...,k-1,1} can be written as a composite of elementary permutations? I thought of starting from {1,2,...,k} then moving the 1 as follows {2,1,3,...,k} -> {2,3,1,4,...,k} and so on until {2,3,...,k-1,1,k} and then doing the same with k "backwards", but switching the 1 and the k isn't an elementary permutation. So what's the right idea?
Thanks for the help!
Give 1 <= i < k, let e_i be the element of S_k (where S_k is the set of all permutations of {1,...,k}) defined by setting e_i (j) = j for j not equal to i,i+1; and e_i (i) = i + 1 and e_i (i + 1) = i.
It goes on to say that if f is in S_k, then it can be written as a composite of elementary permutations. But I don't see how {k,2,3,...,k-1,1} can be written as a composite of elementary permutations? I thought of starting from {1,2,...,k} then moving the 1 as follows {2,1,3,...,k} -> {2,3,1,4,...,k} and so on until {2,3,...,k-1,1,k} and then doing the same with k "backwards", but switching the 1 and the k isn't an elementary permutation. So what's the right idea?
Thanks for the help!