Hints for Homework #2
Chapter 2, #30:
Try the dihedral group D_4 .
Chapter 3, #4:
Apply the definition of the order of an element a as
the least positive integer n such that a^n = e.
If the order of a^-1 is m then (a^-1)^m = e.
Take inverses on both sides of this equation and
try to relate the order of a^-1 to the order of a
in this way.
Chapter 3, #10:
You know that an element a of order 2 satisfies a^2 = e.
What does this say about the inverse of a?
Consider all elements that you could get by multiplying
a and b or their inverses in any sequence. Write down
all the elements that you think you can get and see
if you can show that this subset is a subgroup using
the Two-step Subgroup test.