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.

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

$G≔\mathrm{Symm}\left(4\right)$

$G≔{\mathbf{S}}_4$

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{true}$

$G≔\mathrm{CyclicGroup}\left(6\right)$

$G≔C_6$

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

$\mathrm{true}$

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

$\mathrm{false}$

$\mathrm{IsSemiprimitive}\left(\mathrm{GL}\left(2\,3\right)\right)$

$\mathrm{true}$

$\mathrm{IsQuasiprimitive}\left(\mathrm{GL}\left(2\,3\right)\right)$

$\mathrm{false}$

$\mathrm{andmap}\left(\mathrm{IsTransitive}\mathbf{or}\mathrm{IsSemiRegular}\,\mathrm{remove}\left(\mathrm{IsTrivial}\,\mathrm{NormalSubgroups}\left(\mathrm{GL}\left(2\,3\right)\right)\right)\right)$

$\mathrm{true}$

$\mathrm{andmap}\left(\mathrm{IsTransitive}\,\mathrm{remove}\left(\mathrm{IsTrivial}\,\mathrm{NormalSubgroups}\left(\mathrm{GL}\left(2\,3\right)\right)\right)\right)$

$\mathrm{false}$

$G≔\mathrm{TransitiveGroup}\left(24\,707\right)\:$

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

$\mathrm{true}$

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

$\mathrm{false}$

$\mathrm{map}\left(\mathrm{IsTransitive}\,\mathrm{remove}\left(\mathrm{IsTrivial}\,\mathrm{NormalSubgroups}\left(G\right)\right)\right)$

$\left[\mathrm{true}\,\mathrm{true}\right]$

$G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[4\,5\right]\right]\right)\right]\right)$

$G≔⟨\left(1\,2\,3\right)\,\left(4\,5\right)⟩$

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

$\mathrm{false}$

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

$\mathrm{false}$

$G≔\mathrm{CyclicGroup}\left(72\,'\mathrm{mindegree}'\right)$

$G≔C_{17}$

$\mathrm{orbs}≔\mathrm{map}\left(\mathrm{Elements}\,\mathrm{Orbits}\left(G\right)\right)$

$\mathrm{orbs}≔\left[\left\{1\,2\,3\,4\,5\,6\,7\,8\right\}\,\left\{9\,10\,11\,12\,13\,14\,15\,16\,17\right\}\right]$

$A≔\mathrm{RestrictedPermGroup}\left(G\,\mathrm{orbs}\left[1\right]\right)$

$A≔⟨\left(1\,2\,3\,4\,5\,6\,7\,8\right)⟩$

$\mathrm{IsPrimitive}\left(A\right)$

$\mathrm{false}$

$\mathrm{IsSemiprimitive}\left(A\right)$

$\mathrm{true}$

$\mathrm{IsQuasiprimitive}\left(A\right)$

$\mathrm{false}$