Please USE Pyton Sagemath code to write.Thanks a lot

Problem 1: The 15 puzzle revisited Grading criteria: code correctness for a and b; mathematical correctness for c. Part d is optional. a. Construct a graph I representing the spaces of a 4 x 4 grid, with the vertices labeled A E I M B F J N C G K O D H L P in which two vertices are joined by an edge if the corresponding squares of the grid share a side (not just a corner). b. Form the symmetric group G on the symbols A through P. (Note: for l a list, SymmetricGroup (1) gives the group of permutations acting on 1.) Let S be the subset of G consisting of the transpositions of the form (XY) for each edge {X, Y} in I. Verify that S generates G. Edit C. Find a subset T of S of smallest possible size such that I does in fact generate G. You should verify (computationally, that I generates G, and explain (mathematically) why no smaller set can work. (Hint for the second part: argue in terms of the graph I.) e. Optional: in class, I explained that the 15 puzzle has unsolvable states. For example, starting with the numbers in order (1 at A through 15 at o, with P empty) you cannot arrive at a position where 14 andl4 15 are swapped while everything else remains in place. Explain why this does not contradict anything you did earlier in this problem. Problem 1: The 15 puzzle revisited Grading criteria: code correctness for a and b; mathematical correctness for c. Part d is optional. a. Construct a graph I representing the spaces of a 4 x 4 grid, with the vertices labeled A E I M B F J N C G K O D H L P in which two vertices are joined by an edge if the corresponding squares of the grid share a side (not just a corner). b. Form the symmetric group G on the symbols A through P. (Note: for l a list, SymmetricGroup (1) gives the group of permutations acting on 1.) Let S be the subset of G consisting of the transpositions of the form (XY) for each edge {X, Y} in I. Verify that S generates G. Edit C. Find a subset T of S of smallest possible size such that I does in fact generate G. You should verify (computationally, that I generates G, and explain (mathematically) why no smaller set can work. (Hint for the second part: argue in terms of the graph I.) e. Optional: in class, I explained that the 15 puzzle has unsolvable states. For example, starting with the numbers in order (1 at A through 15 at o, with P empty) you cannot arrive at a position where 14 andl4 15 are swapped while everything else remains in place. Explain why this does not contradict anything you did earlier in this