I'm reading 'A modern treatment of the 15 puzzle' by Archer and cannot understand the proof to Theorem 3 and how they compute the permutation equation with their examples.
My main problem is this computation with regards to the proof: Note that for any permutation $x$ we have $x^{-1}(a_1,\cdots,a_k) x = (a_1x,\cdots,a_kx)$. Thus, $(1,2,\cdots,7)^{-n}(3,4,5)(1,2,...,7)^n$ yields $(1,2,3),\cdots,(5,6,7)$; $(5,6,...,11)^{-n} (7,8,9)(5,6,...,11)^n$ yields $(5,6,7),\cdots,(9,10,11)$ and $(9,10,...,15)^{-n}(11,12,13)(9,10,...,15)^n$ yields $(9,10,11),\cdots,(13,14,15)$
Any help will be greatly appreciated!