Hints and Comments for Section 7.2

NOTE: I highly recommend that you make a real effort to do the problems without using any hints. Only read a hint if you get stuck and remain stuck after making a significant effort.

Problem 11 comment: Here is an example of an infinite group in which every element has finite order. Let H be the cartesian product of an infinite number of copies of ℤ

(a

and where the additions in each component are carried out according to the rules in ℤ

0 = (0, 0, 0, 0, ...).

Now observe that for every element h in H , h + h = 0. This shows that every nonzero element of H has order 2.

Problem 18b hint: Let c = ab. Find and prove an equation for c

Problem 28 hint: One way to get started on a problem like this is to try proving it for small values of the order. For example, try to prove that if |ab| = 2 then |ba| = 2. Then try to prove that if |ab| = 3 then |ba| = 3. If you figure out these special cases you will be in a better position to prove the general finite case: if |ab| = n then |ba| = n. Does your proof apply to elements of infinite order? Does the problem require you to handle the case of elements of infinite order?