Chapter 9.2, Problem 69E

Book Arrangements—A Nonabelian Group Suppose that three books numbered 1, 2, and 3 are placed next to one another on a shelf. If we remove volume 3 and place it before volume 1, the new order of books is 3, 1, 2. Let’s call. We can write

$R=\left(\begin{array}{ccc}1& 2& 3\\ 3& 1& 2\end{array}\right)$

which indicates the books were switched in order from 1, 2, 3 to 3, 1, 2. Other possible replacements are S, T, U, V, and I, as indicated.

$S=\left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\end{array}\right)$ $T=\left(\begin{array}{ccc}1& 2& 3\\ 3& 2& 1\end{array}\right)$ $U=\left(\begin{array}{ccc}1& 2& 3\\ 1& 3& 2\end{array}\right)$

$V=\left(\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right)$ $I=\left(\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\end{array}\right)$

Replacement set I indicates that the books were removed from the shelves and placed back in their original order. Consider the mathematical system with the set of elements {R, S, T, U, V, I} with the operation of *. To evaluate R * S, write

    R * S

$R\ast S=\left(\begin{array}{ccc}1& 2& 3\\ 3& 1& 2\end{array}\right)\ast \left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\end{array}\right)$

As shown in Fig. 9.2, R replaces 1 with 3 and S replaces 3 with 3 (no change), so R * S replaces 1 with 3, R replaces 2 with 1 and S replaces 1 with 2, so R * S

$R\ast S=\left(\begin{array}{ccc}1& 2& 3\\ 3& 2& 1\end{array}\right)=T$

Figure 9.2

replaces 2 with 2 (no change) R replaces 3 with 2 and S replaces 2 with 1, so R * S replaces 3 with 1, R * S replaces 1 with 3, 2 with 2, and 3 with 1.

Since this result is the same as replacement set T, we write R * S = T.

Complete the table for the operation using the procedure outlined.

★	R	S	T	U	V	I
R		T
S
T
U
V
I

b.    Is this mathematical system a group? Explain.

c.     Is this mathematical system a commutative group? Explain.