Group theoretic construction for permutation algorithm Consider a permutation $\sigma = [s_1, \ldots, s_n]$. The `contracting endpoints' construction for the subsequence $[s_i,\ldots, s_k]$ is given by iteratively taking the product of cycles given by the first and last elements of the sequence, successively discarding first and last elements.
Hence, the construction for [2,3,4,5] in [1,2,3,4,5,6] yields (2,5)(3,4).
Can this construction be defined purely in terms of group-theoretic operations?

Assuming you compose permutations right to left, it may be helpful to observe that $(x_2,x_k,x_{k-1},\dots,x_3)(x_1,x_2,\dots,x_k)=(x_1,x_k)$
