The rank of a transitive permutation group $G$ is the number of its sub-orbits, that is, the number of orbits of a point stabilizer.

The PermGroupRank( G ) command returns the rank of the transitive permutation group G. The group G must be a transitive permutation group. If G is not transitive, an exception is raised.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{PermGroupRank}\left(\mathrm{Alt}\left(4\right)\right)$

$2$

$\mathrm{PermGroupRank}\left(\mathrm{Symm}\left(4\right)\right)$

$2$

$S≔\mathrm{Stabilizer}\left(1\,\mathrm{Symm}\left(4\right)\right)$

$S≔⟨\left(3\,4\right)\,\left(2\,3\right)⟩$

$\mathrm{nops}\left(\mathrm{Orbits}\left(S\,\left\{1\,2\,3\,4\right\}\right)\right)$

$2$

$\mathrm{PermGroupRank}\left(\mathrm{FrobeniusGroup}\left(100\,3\right)\right)$

$7$

$G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1\,2\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[3\,4\right]\right]\right)\right]\right)$

$G≔⟨\left(1\,2\right)\,\left(3\,4\right)⟩$

$\mathrm{PermGroupRank}\left(G\right)$

Error, (in GroupTheory:-PermGroupRank) permutation group must be transitive

$\mathrm{IsTransitive}\left(G\right)$

$\mathrm{false}$

The GroupTheory[PermGroupRank] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.