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GroupTheory

 PermGroupRank
 compute the rank of a permutation group

 Calling Sequence PermGroupRank( G )

Parameters

 G - a transitive permutation group

Description

 • The rank of a transitive permutation group $G$ is the number of its sub-orbits, that is, the number of orbits of a point stabiliser. (By convention, the transitivity of an intransitive group is equal to $0$.)
 • 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.

Examples

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

The alternating and symmetric groups of degree $4$ both have rank $2$.

 > $\mathrm{PermGroupRank}\left(\mathrm{Alt}\left(4\right)\right)$
 ${2}$ (1)
 > $\mathrm{PermGroupRank}\left(\mathrm{Symm}\left(4\right)\right)$
 ${2}$ (2)

To verify this, compute the number of orbits of the stabiliser of a point.

 > $S≔\mathrm{Stabiliser}\left(1,\mathrm{Symm}\left(4\right)\right)$
 ${S}{≔}⟨\left({2}{,}{4}\right){,}\left({3}{,}{4}\right){,}\left({2}{,}{4}{,}{3}\right){,}\left({2}{,}{3}{,}{4}\right)⟩$ (3)
 > $\mathrm{nops}\left(\mathrm{Orbits}\left(S\right)\right)$
 ${2}$ (4)
 > $\mathrm{PermGroupRank}\left(\mathrm{FrobeniusGroup}\left(100,3\right)\right)$
 ${7}$ (5)

Note that the rank is only defined for transitive permutation groups. If the input to the PermGroupRank command is an intransitive group, then an exception is raised, as in the following example.

 > $G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right),\mathrm{Perm}\left(\left[\left[3,4\right]\right]\right)\right]\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({3}{,}{4}\right)⟩$ (6)
 > $\mathrm{PermGroupRank}\left(G\right)$
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{false}}$ (7)

Compatibility

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