HINTS

To compute a sign use the disjoint cycles notation for a permutation.

Use injectivity of g.

First count the number of 3-cycles disjoint from a given 2-cycle.

You may want to use the permutation calculator.

1. Show that multiplication by a is injective using ax = ay mod n implies x = y.
2. Use: if x and y are invertible mod n then xy is invertible mod n.

What would be the inverse of multiplication by a?

1. Distinguish the cases 1) i supp(g), 2) i supp(h), 3) i belongs to neither supp(g), nor supp(h).
2. Interchange factors repeatedly in the product ghgh ··· .
3. Distinguish the cases 1) i supp(g), 2) i supp(h), 3) i belongs to neither supp(g), nor supp(h).
4. Apply division with remainder to m and t.
5. If m = lcm(m₁, m₂, ..., m_r), show that g^m = id. Then show that if g^p = id, the number p is divisible by m₁, m₂, ..., m_r.

1. Try to write (1,2) as a product of three 4-cycles of S₄. Then use conjugation to check that each transposition is a product of 4-cycles. Conclude that each permutation is a product of 4-cycles.

2. Try to write (1,2,3) as a product of two 5-cycles of S₅. Then use conjugation to check that each 3-cycle is a product of 5-cycles. Conclude that each even permutation is a product of 5-cycles.

Use the conjugation formulae.

1. Look at g(i,j)g^-1.
2. Use 1.
3. Use 2.

Determine first which cycle structures occur among elements in A₄.

Can you write each permutation with a and b?

1. Do you see a relation with switching rows or columns?
2. Compare the entries of the permutation matrix of g with those of MN. How many nonzero entries does a row (or column) of a permutation matrix have?
3. What happens to the determinant of a matrix if you switch two of its columns?
4. Use that both sgn and det are multiplicative.

Make a picture.

Use the definition of determinant, where the determinant is written as a sum over n! terms. Relate entries A_i,g(i) and A_g^-1(k),k and then compare termwise the expression for det(A) and det(A^T). Use sgn(g) = sgn(g^-1) and that g^-1 ranges through S_n if g does.

1. List the elements.
2. Produce a few permutations as in Exercise 12.
3. Consider what happens to the set R consisting of the three rows and the set C consisting of the three columns when you apply a single move and a composition of moves. Use this to bring S₃ × S₃ into the picture.
4. Just experiment and see how far you get.

1. For each face there are two moves.
2. Analyse what happens with each vertex i and its new label g(i).
3. What is the sign of each move?