Hints for Homework Assignment 8 5.19: If there were an odd permutation alpha in H, what happens if you multiply H by alpha? You get H again. (Why?) What happens to an even or odd permutation when you multiply by alpha? 5.28: Write beta in disjoint cycle notation and then apply Theorem 5.3. 6.2: As we saw in Example 13, alpha(k) = k alpha(1). What are the possible choices for alpha(1) that give you an isomorphism? (i.e., one-to-one, onto, as well as satisfying the homomorphism property). 6.6: Compose the isomorphisms and show that the composition satisfies the properties of an isomorphism: one-to-one, onto, and the homomorphism property. To show the homomorphism property for the composition, write out the definition and apply the homomorphism property for both isomorphisms in the composition. 6.7: Check out Gallian's advice on this topic, which we discussed in class. What is the largest possible order for an element in each group? 6.10: alpha(ab) = (ab)^{-1} by definition, and alpha(ab) = alpha(a) alpha(b) by the homomorphism property.