In S₆ we choose the permutations a = (1,2,3), b = (2,3,4,5,6) and c = (1,4,6,3).

1. Calculate a^-1, abc, abc², c^-1b and (acb)^-1.
2. Calculate the sign of each of the above permutations.

Let g be a permutation in S_n. Show that if i supp(g), then g(i) supp(g).

How many elements of S₅ have the cycle structure 2,3?

Let g be the permutation (1,2,3)(2,3,4)(3,4,5)(4,5,6)(5,6,7)(6,7,8)(7,8,9) in S₉.

1. Write g as a product of disjoint cycles.
2. Calculate the fixed points of g.
3. Write g^-1 as a product of disjoint cycles.
4. Is g even?

Let n be an integer greater than 2. Suppose a has an inverse mod n. Label the elements of the set S of residue classes 1, 2, ..., n - 1 in Z/nZ in the obvious way with the integers 1, 2, ..., n -1.

1. Show that multiplication by a defines a permutation p_a of 1, 2, ..., n - 1. For a = 2 and n = 9 write the corresponding permutation as a product of disjoint cycles. Can you read off the smallest positive integer m such that a^m = 1 mod n?
2. Suppose p_a is written as a product of disjoint cycles. Prove that the cycles fall into two categories: One consisting of cycles all of whose entries are invertible mod n and one consisting of cycles all of whose entries are not invertible mod n.

Let R be the residue class ring Z/3Z[X]/(X² + 1) and let a be the class of X. Then a is an invertible element of R. Show that multiplication by a produces a permutation of these elements. Write this permutation as a product of disjoint cycles. What is its cycle structure?

1. If the permutations g and h in S_n have disjoint supports, then g and h commute, i.e., gh = hg. Prove this.
2. Suppose that the permutations g and h in S_n commute. Prove that (gh)^m = g^mh^m for all positive numbers m.
3. Suppose that the permutations g and h in S_n have disjoint supports. Prove that (gh)^m = id for some positive number m implies that g^m = id and h^m = id.
4. If the permutation has order t and if g^m = id for some positive number m, show that t divides m. In particular, if c is a t-cycle and c^m = id for some positive number m, then m is divisible by t.
5. Prove that if the permutation g has cycle structure m₁, m₂, ..., m_r, then the order of g equals lcm(m₁, m₂, ..., m_r).

1. Prove that for n > 4 every permutation in S_n can be written as a product of 4-cycles.

2. Prove that for n > 5 every even permutation can be written as a product of 5-cycles.

Let a = (1,2,3)(4,7,9)(5,6). Determine an element b S₉ such that bab^-1 = (9,8,7)(6,5,4)(3,2).

Let g be an element of S_n with n > 2.

1. If g commutes with the transposition (i,j), where i j, then g(i) {i,j}. Prove this.
2. Show that g(i) = i, whenever g commutes with the transpositions (i,j) and (i,k), where i, j, k are mutually distinct.
3. Prove that the identity map is the only permutation in S_n that commutes with all elements of S_n.

Write all elements of A₄ as products of disjoint cycles.

Let a = (1,2) and b = (2, ..., n).

1. Calculate bab^-1.
2. Calculate b^kab^-k, for k N.
3. Prove that every element of S_n can be written as a product of elements from {a, b, b^-1}.

For g S_n, we define a matrix M by

The matrix M is called the permutation matrix of g.

1. Calculate the permutation matrices for the 6 permutations of S₃.
2. Prove: If g, h S_n with associated permutation matrices M and N, then the permutation matrix of gh is MN.
3. Prove: If g is a transposition, then det(M) = -1.
4. Show that sgn(g) = det(M).

Label the vertices of a quadrangle with the numbers 1 to 4.

1. Which permutation of the four vertices describes the rotation through +90° whose center is the middle point of the quadrangle? And which one describes the reflection in the diagonal through the vertices 1 and 3?
2. Determine the permutations g of S₄ satisfying: If {i, j} is an edge of the quadrangle, then so is {g(i), g(j)}.
3. Describe each of the permutations of the above part in geometric terms as a reflection or a rotation. Which of these permutations are even?

Prove that the determinant of a square matrix A and the determinant of its transpose A^T are equal, i.e., prove

Put the numbers 1, 2, 3, 4 into a 2 by 2 matrix as follows. Put 1 on spot 11, 2 on spot 12, 3 on spot 21 and 4 on spot 22, i.e., the matrix [[1, 2], [3, 4]].

1. Suppose you are allowed to interchange two columns or two rows. Which permutations of S₄ can you get using these moves repeatedly? What if you allow as extra type of move a reflection in the diagonal of the matrix?
2. Suppose you are allowed to do the following types of moves: Choose a column or row and interchange the two entries. What permutations do you get this way?
3. Now consider the 3 by 3 matrix [[1,2,3], [4,5,6], [7,8,9]]. Individual moves are: Choose two rows (or two columns) and interchange them. Show that you can label each resulting permutation with a pair of permutations from S₃ × S₃. Conclude that you get 36 permutations.
4. Experiment with a 3 by 3 matrix, where a single move consists of shifting the entries of an individual column or row cyclically. With the techniques of Chapter 8, you will be able to deal with such problems effectively.

Label the vertices of a regular tetrahedron with the integers 1, 2, 3, 4 (see figure).

Consider the following moves: For each face of the tetrahedron the corresponding move consists of turning the face 120 degrees clockwise or counter clockwise and moving the labels accordingly (so the vertex opposite the face remains fixed). After applying a number of moves, we read off the resulting permutation g in the obvious way: g(i) is the new label of vertex i.

1. List the 8 moves as permutations.
2. Suppose, after a number of moves, we have obtained the permutation g. Show that applying a move h leads to the permutation gh^-1.
3. Which permutations of 1, 2, 3, 4 can you get by using these moves?