Mathematics 243 – Algebraic Structures Discussion 3 – Examples of Groups November 8, 2006; Writeups due: Monday, November 13

Background

In the last two classes, we have introduced the definition of a group and some first examples. Today we want to continue and study some additional properties and examples of groups of various kinds.

Discussion Questions

I. Let G be the set of six 3 × 3 matrices given below:

\begin{aligned}I = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , S & = ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) , T = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} ) , \\ U = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) , V & = ( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ) , W = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) . \end{aligned}

G is a group under the operation of matrix multiplication. Give the operation table, labeling the rows and columns in this order. (Write the entries of the table using the names I, S, T, U, V, W, not the matrices!)

II. Let K be the set of six functions (with domain R − {0, 1}):

K = \{f_{1}(x) = x, f_{2}(x) = \frac{1}{1 − x}, f_{3}(x) = \frac{x − 1}{x}, f_{4}(x) = \frac{1}{x}, f_{5}(x) = 1 − x, f_{6}(x) = \frac{x}{x − 1}\} .

K is a group under the operation of function composition. Give the operation table, labeling the rows and columns by the functions in this order.

III. The 2 × 2 matrix A given by

A = ( \begin{matrix}\frac{1}{2} & \frac{−\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{matrix} )

is an invertible matrix (why?). What is the smallest group H of invertible 2 × 2 matrices containing A? Hint: A group must be “closed” under matrix multiplication – multiplying any two matrices in H must produce a matrix in the same set. So in particular, A · A = A^{2}, A^{3}, …, A^{−1}, … must also be in H. Give the group operation table of H.

over for question IV A) Comparing the example groups G, H, and K from questions I, II, and III, if you look

closely, you should start to realize that two of these are “the same”. In what way? What should mean for two groups to be “the same”? Recall that we saw in class on Monday that just saying the two groups have the same order (same number of elements) is not sufficient.

B) Each of the groups in questions I,II,III contains several (proper) subsets that are groups in their own right under the given operation. Find at least three such subsets in each case.