Mathematics 243 – Algebraic Structures Discussion 3 – Examples of Groups October 31, 2003; Writeups due: Friday, November 7

Background

In Wednesday’s class, we 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

A) Show that if (gh)^{−1} = g^{−1}h^{−1} for all g, h in a group G, then G is abelian (that is the

group operation is commutative). B) 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!)

C) 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 its operation – combining any two elements using the operation must produce an element of the same set. So in particular, A · A = A^{2} must also be in H. Give the group operation table of H.

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

K = \{x, \frac{1}{1 − x}, \frac{x − 1}{x}, \frac{1}{x}, 1 − 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. It may help to abbreviate the functions by giving them names(!)

E) Comparing the example groups G, H, and K from questions B, C, and D, 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 Wednesday, that just saying the two groups have the same order is not sufficient.