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.