Hints for Homework 5

Here are some hints for exercises in Chapter 4.

18: Write the cyclic group with generator a as <a>.
If this group contains an element of infinite order,
what can you say about the order of generator a?

25: Note that the rotations form a cyclic subgroup,
allowing you to apply Theorem 4.4.  When counting
elements of order 2, in addition to the reflections,
determine when R_{180} is a symmetry.  It may help
when thinking about this problem to know that
F R = R^{-1} F  (and R F = F R^{-1}).  Feel free
to cite this fact if you think it will help.
You can prove this with an argument similar to
Exercise 7 in Chapter 3.

49: Let a, b be elements with |a| = 4, |b| = 5.
What is |ab|?  Remember that the group is
Abelian.