# Primitivity Properties on Coset Geometries

## `IsPrimitive(C): CosetGeom -> BoolElt`

## `IsPRI(C): CosetGeom -> BoolElt`

## `IsPrimitive(C): CosetGeom -> BoolElt`

## `IsPRI(C): CosetGeom -> BoolElt`

Given a coset geometry $C$, this function returns the boolean value `true` if and only if $C$ is a primitive geometry, i.e. all of its maximal parabolic subgroups are maximal subgroups of its group.

## `IsWeaklyPrimitive(C): CosetGeom -> BoolElt`

## `IsWPRI(C): CosetGeom -> BoolElt`

## `IsWeaklyPrimitive(C): CosetGeom -> BoolElt`

## `IsWPRI(C): CosetGeom -> BoolElt`

Given a coset geometry $C$, this function returns the boolean value `true` if and only if $C$ is a weakly primitive geometry, i.e. at least one of its maximal parabolic subgroups is a maximal subgroup of its group.

## `IsResiduallyPrimitive(C): CosetGeom -> BoolElt`

## `IsRPRI(C): CosetGeom -> BoolElt`

## `IsResiduallyPrimitive(C): CosetGeom -> BoolElt`

## `IsRPRI(C): CosetGeom -> BoolElt`

Given a coset geometry $C$, this function returns the boolean value `true` if and only if $C$ is a primitive geometry and all of its residues are as well.

## `IsResiduallyWealyPrimitive(C): CosetGeom -> BoolElt`

## `IsRWPRI(C): CosetGeom -> BoolElt`

## `IsRWP(C): CosetGeom -> BoolElt`

## `IsResiduallyWeaklyPrimitive(C): CosetGeom -> BoolElt`

## `IsRWPRI(C): CosetGeom -> BoolElt`

## `IsRWP(C): CosetGeom -> BoolElt`

Given a coset geometry $C$, this function returns the boolean value `true` if and only if $C$ is a weakly primitive geometry and all of its residues are as well.

## `IsLocallyTwoTransitive(C): CosetGeom -> BoolElt`

## `Is2T1(C): CosetGeom -> BoolElt`

Given a coset geometry $C$, this function returns the boolean value `true` if and only if $C$ is locally two-transitive, i.e. all of its minimal parabolic subgroups have a two-transitive action on the cosets of the Borel subgroup.

## `LocallySArcTransitive(C): CosetGeom -> RngIntElt`

Given a coset geometry $C$, this function returns the largest integer $s$ such that $C$ is locally $s$-arc-transitive but not locally $(s+1)$-arc-transitive. If the function returns 10, it means $C$ is a coset geometry corresponding to a polygon and hence this value must be considered as infinity.