permgroup - Maple Programming Help

permgroup

represent a permutation group

 Calling Sequence permgroup(deg, gens)

Parameters

 deg - degree of the permutation group gens - set of generators for the permutation group

Description

 • The function permgroup is used as a procedure and an unevaluated procedure call. As a procedure, permgroup checks its arguments and then either exits with an error or returns the unevaluated permgroup call.
 • The first argument is the degree of the group, and should be an integer. The second argument is a set of group generators. Each generator is represented in disjoint cycle notation. The generators may be named or unnamed. A named generator is an equation; the left operand is the generator's name, the right operand is the permutation in disjoint cycle notation.
 • A permutation in disjoint cycle notation is a list of lists. Each sub-list represents a cycle; the permutation is the product of these cycles. The cycle $[{a}_{1},{a}_{2},...,{a}_{n}]$ represents the permutation which maps ${a}_{1}$ to ${a}_{2}$, ${a}_{2}$ to ${a}_{3}$, ..., ${a}_{n-1}$ to ${a}_{n}$, and ${a}_{n}$ to ${a}_{1}$. The identity element is represented by the empty list $\left[\right]$.
 • The permgroup function follows the convention that permutations act on the right''. In other words, if $\mathrm{p1}$ and $\mathrm{p2}$ are permutations, then the product of $\mathrm{p1}$ and $\mathrm{p2}$, $\mathrm{p1}&*\mathrm{p2}$ is defined such that $\mathrm{p1}&*\mathrm{p2}\left(i\right)=\mathrm{p2}\left(\mathrm{p1}\left(i\right)\right)$ for $i=1..\mathrm{deg}$.

Examples

 > $\mathrm{permgroup}\left(5,\left\{a=\left[\left[1,2\right],\left[4,5\right]\right],b=\left[\left[5,4,3,2,1\right]\right]\right\}\right)$
 ${\mathrm{permgroup}}{}\left({5}{,}\left\{{a}{=}\left[\left[{1}{,}{2}\right]{,}\left[{4}{,}{5}\right]\right]{,}{b}{=}\left[\left[{5}{,}{4}{,}{3}{,}{2}{,}{1}\right]\right]\right\}\right)$ (1)
 > $\mathrm{permgroup}\left(6,\left\{\left[\left[1,2\right]\right],\left[\left[1,2,3,4,5,6\right]\right]\right\}\right)$
 ${\mathrm{permgroup}}{}\left({6}{,}\left\{\left[\left[{1}{,}{2}\right]\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}\right]\right]\right\}\right)$ (2)

the following is not legal:

 > $\mathrm{permgroup}\left(5,\left\{x=\left[\left[3,4\right]\right],y=\left[\left[7,2\right]\right]\right\}\right)$