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.