Important: The permgroup command has been deprecated. Use the superseding command GroupTheory[Group] instead.

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 $\left(\mathrm{p1}\&*\mathrm{p2}\right)\left(i\right)=\mathrm{p2}\left(\mathrm{p1}\left(i\right)\right)$ for $i=1..\mathrm{deg}$.

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

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

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

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

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

Error, (in permgroup) generators must represent products of disjoint cycles, but [[7, 2]] does not