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.

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

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

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

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

$\mathrm{false}$

$\mathrm{IsPrimitive}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsPrimitive}\left(\mathrm{PGU}\left(3\,3\right)\right)$

$\mathrm{true}$

$\mathrm{IsPrimitive}\left(\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{IsPrimitive}\left(\mathrm{DihedralGroup}\left(5\right)\right)$

$\mathrm{true}$

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

For more information on Maple 17 changes, see Updates in Maple 17.