What is the order of the permutation 1348679 in s9 how many

Question

Assignment 5-

1. (a) What is the order of the permutation (13)(486)(79) in S₉? How many permutations have the same cycle type as it?

(b) Determine the disjoint cycle decomposition of the permutation

(12)(23)(34)(45)(34)(23)(12).

(c) Write the 7-cycle (1234567) as a product of the transpositions (12),(13),(14),(45),(56),(57).

2. Show that every even permutation can be written as the product of 3-cycles (with possible repetitions).

3. Let A be an n × n matrix. The determinant of A can be defined in terms of permutations as follows:

det(A) = ∑_{σ∈S_n}sgn(σ)A_1,σ(1)A_2,σ(2) · · · A_n,σ(n).

(a) Use the definition above to show that exchanging any two rows of a matrix changes its determinant by a factor of -1.

(b) Use the definition above to show that det(A) = det(A^T).

4. Consider the action of the symmetry group of the object below acting on its vertices. What is the size of the orbit of a vertex? What is the size of the stabilizer of a vertex?

5. A coloring of the grid below is an assignment of colors to its triangular cells, where any given triangular cell is colored red or green. Determine the number of colorings of the grid, with the condition that two colorings are considered the same if one can be obtained from the other by rotating the grid about its center.