Hints for Homework 6
Chapter 4
9: Be careful to distinguish between additive and multiplicative
notation.
46: Apply Theorem 4.2.
52: Think of this as a cyclic group of order 42. Then
apply Corollary 2 of Theorem 4.2.
Chapter 5
3: Apply Theorem 5.3. Note that some of the permutations are written
as a product of distinct cycles, and some are not.
8: Consider an element of A_{10} as a permutation written in disjoint
cycle notation. The lengths of the cycles must add up to no more than
10, since the permutations are of degree 10. Odd cycles have lengths
2, 4, 6, 8. Even cycles have lengths 3, 5, 7, 9. (Justify these
assertions, cf. Exercise 5.11.) Since we're dealing with the
alternating group, odd cycles must occur in pairs, otherwise you would
have an odd permutation (not an even one). Determine the combination
of cycle lengths that add up to no more than 10, form an even
permutation, and have the largest LCM.
12: You can argue two ways. First way: Consider alpha * alpha^{-1}
and apply the Lemma on page 101. Second way: write alpha as a product
of an even number of transpositions and then explicitly represent its
inverse.