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
hint if you get stuck and remain stuck after making a significant
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 (a1,
a2, a3, a4, ... )
all of whose entries are 0 or 1. Define an addition operation on H by adding two sequences component
(a1, a2, a3, a4, ... )
+ (b1, b2, b3, b4, ... )
= (a1 + b1, a2 + b2, a3 + b3, a4 + 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
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
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?