Hints for Homework 10 Chapter 6: 35, 36, 40 35: D_4 works. The only possible orders for elements in D_4 are 1, 2, and 4. Look for an element of order 4 for which the corresponding inner automorphism has order 2. Be careful to compose the inner automorphism with itself as a function, not as if it were a group element of D_4. 36: Consider the map phi that sends 2n to 3n (for any integer n). Verify all the properties to show that this map is an isomorphism. The homomorphism property for addition is phi(2m + 2n) = phi(2m) + phi(2n). Determine whether this map satisfies the homomorphism property for multiplication: phi( 2m * 2n ) = phi(2m) * phi(2n). 40: Let a, b be positive integers. Consider phi(a/b) = phi(1/b) + ... + phi(1/b), repeated a times. Consider phi(1) = phi(1/b) + ... + phi(1/b), repeated b times. Chapter 7: 4, 7, 8, 10, 15, 16, 17, 22 10: Use Lagrange's theorem to determine the possible orders of elements of G. Consider the subgroup generated by two nonidentity elements a and b of different orders and apply Lagrange's Theorem again to decide what possible orders this subgroup could have such that its order satisfies Lagrange's theorem, given that it contains both a and b. 15: Think Lagrange's Theorem and Corollary 3. 16: Think Corollary 4. 22: Since G has more than one element, select any non-identity element a in G. Rule out the infinite case. Consider and apply Lagrange's Theorem. What can the order of a be? What does that tell us about G? Chapter 8: 1 1: I described the identity element and the inverse of an element in a direct product in lecture. Use this to verify that the direct product satisfies all properties needed to be a group. All you need to do is relate all properties to the component group properties.