GroupTheory/IsSemiprimitive - Maple Help

GroupTheory

 IsSemiprimitive
 determine whether a permutation group is semi-primitive
 IsQuasiprimitive
 determine whether a permutation group is quasi-primitive

 Calling Sequence IsSemiprimitive( G, domain ) IsQuasiprimitive( G, domain )

Parameters

 G - : PermutationGroup : a permutation group domain - : set(posint) : (optional) a G-invariant subset of the support of G

Description

 • A permutation group $G$ is quasi-primitive if each of its non-trivial normal subgroups is transitive.
 • A permutation group $G$ is semi-primitive if each of its non-trivial normal subgroups either is transitive or semi-regular.
 • Because every non-trivial normal subgroup of a primitive permutation group is transitive, it is clear that semi-primitivity and quasi-primitivity are generalizations of primitivity. In particular, every primitive permutation group is both semi-primitive and quasi-primitive. It also follows from the definitions that a quasi-primitive permutation group is semi-primitive.
 • The IsQuasiprimitive( G ) command returns true if the permutation group G is quasi-primitive, and returns the value false otherwise.
 • The IsSemiprimitive( G ) command returns true if the permutation group G is semi-primitive, and returns false if it is not.
 • The optional domain argument, which must be a G-invariant set, can be used to specify a particular domain of action for G. By default, domain is equal to the support of G, that is, the set of points displaced by some element of G.

Examples

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

Since symmetric groups are primitive, they are also both semi-primitive and quasi-primitive.

 > $G≔\mathrm{Symm}\left(4\right)$
 ${G}{≔}{{\mathbf{S}}}_{{4}}$ (1)
 > $\mathrm{IsSemiprimitive}\left(G\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{IsQuasiprimitive}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{true}}$ (4)

The cyclic group of order $6$ is semi-primitive, but not quasi-primitive.

 > $G≔\mathrm{CyclicGroup}\left(6\right)$
 ${G}{≔}{{C}}_{{6}}$ (5)
 > $\mathrm{IsSemiprimitive}\left(G\right)$
 ${\mathrm{true}}$ (6)
 > $\mathrm{IsQuasiprimitive}\left(G\right)$
 ${\mathrm{false}}$ (7)
 > $\mathrm{IsSemiprimitive}\left(\mathrm{GL}\left(2,3\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{IsQuasiprimitive}\left(\mathrm{GL}\left(2,3\right)\right)$
 ${\mathrm{false}}$ (9)
 > $\mathrm{andmap}\left(\mathrm{IsTransitive}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{or}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{IsSemiRegular},\mathrm{remove}\left(\mathrm{IsTrivial},\mathrm{NormalSubgroups}\left(\mathrm{GL}\left(2,3\right)\right)\right)\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{andmap}\left(\mathrm{IsTransitive},\mathrm{remove}\left(\mathrm{IsTrivial},\mathrm{NormalSubgroups}\left(\mathrm{GL}\left(2,3\right)\right)\right)\right)$
 ${\mathrm{false}}$ (11)
 > $G≔\mathrm{TransitiveGroup}\left(24,707\right):$
 > $\mathrm{IsQuasiprimitive}\left(G\right)$
 ${\mathrm{true}}$ (12)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{false}}$ (13)
 > $\mathrm{map}\left(\mathrm{IsTransitive},\mathrm{remove}\left(\mathrm{IsTrivial},\mathrm{NormalSubgroups}\left(G\right)\right)\right)$
 $\left[{\mathrm{true}}{,}{\mathrm{true}}\right]$ (14)

The following groups fail to be semi-primitive (hence, also quasi-primitive) since they are not even transitive.

 > $G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1,2,3\right]\right]\right),\mathrm{Perm}\left(\left[\left[4,5\right]\right]\right)\right]\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}\right){,}\left({4}{,}{5}\right)⟩$ (15)
 > $\mathrm{IsSemiprimitive}\left(G\right)$
 ${\mathrm{false}}$ (16)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{false}}$ (17)
 > $G≔\mathrm{CyclicGroup}\left(72,'\mathrm{mindegree}'\right)$
 ${G}{≔}{{C}}_{{17}}$ (18)
 > $\mathrm{orbs}≔\mathrm{map}\left(\mathrm{Elements},\mathrm{Orbits}\left(G\right)\right)$
 ${\mathrm{orbs}}{≔}\left[\left\{{1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right\}{,}\left\{{9}{,}{10}{,}{11}{,}{12}{,}{13}{,}{14}{,}{15}{,}{16}{,}{17}\right\}\right]$ (19)
 > $A≔\mathrm{RestrictedPermGroup}\left(G,\mathrm{orbs}\left[1\right]\right)$
 ${A}{≔}⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right)⟩$ (20)
 > $\mathrm{IsPrimitive}\left(A\right)$
 ${\mathrm{false}}$ (21)
 > $\mathrm{IsSemiprimitive}\left(A\right)$
 ${\mathrm{true}}$ (22)
 > $\mathrm{IsQuasiprimitive}\left(A\right)$
 ${\mathrm{false}}$ (23)