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 ℤ2.  Thus the elements of H are infinite sequences of the form (a1a2a3a4,  ... )  all of whose entries are 0 or 1.  Define an addition operation on H by adding two sequences component wise: 
                        (a1a2a3a4,  ... )  +  (b1b2b3b4,  ... )  =  (a1 + b1a2 + b2a3 + b3a4 + b4,  ... )
and where the additions in each component are carried out according to the rules in ℤ2.  This is a group, as you can verify directly from the definition of group, and the identity element is the sequence
= (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 cn .

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?