Book Arrangements Suppose that three books numbered 1, 2,

Chapter 10, Problem 10.1.121

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Book Arrangements 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 this replacement R. We can write

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

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.

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

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

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

As shown in Fig. 10.2, 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 \(R * S\) 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.

a) Complete the table for the operation using the procedure outlined.

b) Is this mathematical system a group? Explain.

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