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 <a,b> 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 <a> 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.