Problem 1. Prove that the conjugacy classes for the group are given by
Problem 2. More generally, let be a group, and suppose are conjugate to each other (i.e. that there exists so that ).
Prove that if and only if .
Note: This is the key step in understanding the conjugacy classes of any group. The conjugacy class of a transposition must contain only elements of order 2 which therefore must be other transpositions. Similarly, the conjugacy class of a 3-cycle must contain only 3-cycles.
Problem 3. We developed a “natural” representation of wherein each element is associated with a “permutation matrix.” In particular, this matrix, is defined by
Unfortunately, this representation is reducible. Indeed, every such matrix leaves the following subspace invariant:
As a result, the “complementary subpsace” , defined by
is also left invariant by every matrix . We argued that every matrix acts on the subspace as the identity function, i.e. that as a matrix acting on the 1-dimensional space we have
as a transformation .
The piece of our 3-dimensional representation we obtain by restricting to gives us the trivial irreducible representation, one that we already had. Hence we focus on representing the permutations as matrices .
Using the basis we found that
This means that as a linear transformation on , the element can be represented by the matrix
Compute the other five matrices, for every .
Note: We should expect that if and are in the same conjugacy class, then their matrix representatives and will look different but will have the same trace.
Problem 4. Complete the character table for . (This should be easy after finishing problem 3.)
Problem 5. Suppose is a finite Abelian group of size . Prove that must have conjugacy classes. What does this result tell us about how simplified a character table is for an Abelian group?
Problem 6. Complete the character table for the group . Do this by following this step-by-step procedure.
Step (1) : Figure out how large the table is by counting the number of conjugacy classes (Problem 5. should be helpful.)
Step (2) : Include as many “obvious” or “easy” irreducible representations as you can. For instance, the trivial irreducible representation
should be used to fill out the first row. For remaining rows (assuming there are any), see if you can think of any “natural” transformations that the elements in remind you of.
Step (3) Cry? Try and invent new representations?? Look things up? Consult a Chemistry textbook with an example worked out where is the symmetry group???