Permutations

In GAP, to obtain the permutation matrix $P$ corresponding to the cyclic permutation of $\mathbb{Z}_n$ which sends $a_1\longmapsto a_2$, $a_2\longmapsto a_3$, ..., $a_{n-1}\longmapsto a_n$, and $a_n\longmapsto a_1$, type P:=PermutationMat(($a_1$,...,$a_n$),n);. For example, to obtain the permutation matrix $P$ corresponding to the permutation of $\mathbb{Z}_5$ which sends $1\longmapsto 2$, $2\longmapsto 3$, $3\longmapsto 5$, and $5\longmapsto 1$, type P:=PermutationMat((1,2,3,5),5);. This is printed out as a list of row vectors. To get $P$ to print out as a square matrix, type PrintArray(P). To get the inverse permutation, type PrintArray(P^(-1));.

a^(-1) is the inverse. For example, if you type

a:=(1,2,3,5);
b:=(3,2,1)(4,5)
a*b;
a^(-1);

returns the product (3,4,5) and the inverse (1,5,3,2).

Exercise 5.19.1 (a)   Compute $(1,2,3,4,5,6,7,8,9)^3(1,2,3,4,5)^{-5}$.

(b) Make a $6\times 6$ table of all $36$ products $ab$, where $a,b$ run over all $6$ permutations of $\{1,2,3\}$.