IsPrimitive - Maple Help

GroupTheory

 IsPrimitive
 determine whether a permutation group is primitive

 Calling Sequence IsPrimitive( G )

Parameters

 G - a permutation group

Description

 • A block for a permutation group $G$, acting on a set $\mathrm{\Omega }$ , is a subset $B$ of $\mathrm{\Omega }$ such that, for all $g$ in $G$, either ${B}^{g}=B$ or ${B}^{g}$ and $B$ are disjoint. A block $B$ is trivial if it consists of a single point or if $B=\mathrm{\Omega }$ . A transitive permutation group $G$ is primitive if it possesses no non-trivial block. Note that an intransitive group is not primitive.
 • The IsPrimitive( G ) command returns true if the permutation group G is primitive, and returns false otherwise. The group G must be an instance of a permutation group.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{PermutationGroup}\left(\left\{\left[\left[1,2\right]\right],\left[\left[1,2,3\right],\left[4,5\right]\right]\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (1)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsPrimitive}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsPrimitive}\left(\mathrm{PGU}\left(3,3\right)\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsPrimitive}\left(\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{false}}$ (5)
 > $\mathrm{IsPrimitive}\left(\mathrm{DihedralGroup}\left(5\right)\right)$
 ${\mathrm{true}}$ (6)

Compatibility

 • The GroupTheory[IsPrimitive] command was introduced in Maple 17.