These homework problems are meant to expand your understanding of what goes on during class. Any you turn in will be graded and returned to you. Answers may or may not be posted on the web, depending on demand.

1. The one part of Lagrange's theorem we didn't prove is the following lemma: Suppose H is a subgroup of a finite group G. Then, for any element g of G, the order of gH equals the order of H. Let's prove this as follows: first, notice that |gH| <= |H|. Next, show that |gH| >= |H| by showing that one can't have gh₁ = gh₂ for two distinct elements h₁ and h₂ of H. (Sorry, I've used "<=" and ">=" as short-hand for "less than or equal to" and "greater than or equal to" due to local font problems.)

2. Complete a Cayley table (that is, a multiplication table) for D₃ and D₄, the sets of symmetries of an equilateral triangle and square. Use the notation that D_n = {1,r,...,r^n-1,m,mr,...,mr^n-1}, where r is a minimal counterclockwise rotation and m is a mirror reflection, as in class. (This is mostly so you'll have these handy for the next two problems.)

3. Consider the element m in D₃. We say that an element y in D₃ is conjugate with m if y=x^-1mx for some x in D₃. So mr² is conjugate with m since r^-1mr = mr².
   1. Find all elements in D₃ that are conjugate with m. We call this set of elements the conjugacy class of m.

   2. Find all other conjugacy classes in D₃.

   3. Do any of your conjugacy classes intersect?

   4. Does any element not belong to a conjugacy class.

4. Find all the conjugacy classes of D₄. Can you determine the conjugacy classes of D_n?

5. Prove the following facts for any group, G, and elements x, y and z in G.
   1. If x is an element of G then x is conjugate with itself.

   2. If x and y are conjugate, then y and x are conjugate.

   3. If x and y are conjugate and y and z are conjugate then x and z are conjugate.

   These properties are called the reflexive, symmetric and transitive properties. Any relation, such as conjugacy, which possesses these properties is said to be an equivalence relation.

6. Write a short computer program to determine all possible vertex types. Recall that a vertex type is an integer solution {n₁,n₂, ... ,n_k} of the equation
   

   n1 - 2 n1 + ... +  nk - 2 nk = 2

   or, equivalently,
   

   ( (  1 2 -  1 n1 ) ) + ... + ( (  1 2 -  1 nk ) ) = 1.

   (Please excuse the kludged parentheses.) (The n_j are all at least 3, and may repeat values.) You may assume what we proved in class: there are no solutions with any n over 42.

These problems are also available as a PDF file.