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.

A block system for $G$ is a collection of blocks that are the images of one of its members; that is, the orbit of a block under the induced action of $G$ on the subsets of $\mathrm{\Omega }$.

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.

You can pass an optional second domain argument to check whether G acts primitively on the subset domain of its support. By default, domain is the entire support of G.

The BlockSystem( G ) command returns a non-trivial block system for G if one exists or, in case G is primitive, the block system consisting only of $\mathrm{\Omega }$.

If the optional 'containing' = S option is passed, then the BlockSystem command returns a non-trivial block system, provided that one exists, in which the subset S of the domain of G is contained entirely within one block. The resulting block system is such that the blocks are minimal with respect to including the set S within a single block.

The BlocksImage( G, B ) command returns a permutation group permutation equivalent to the action of $G$ on the block system $B$.

The MinimalBlockSystem( G ) command returns a minimal block system for G provided one exists. The members of the block system are maximal blocks with respect to inclusion. If G is primitive, then the trivial block system consisting of the entire support of G is returned.

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

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

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

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

$\mathrm{false}$

$\mathrm{IsPrimitive}\left(\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1\,2\,4\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[4\,5\,7\right]\right]\right)\right\}\right)\right)$

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{false}$

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

$\mathrm{true}$

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

$\mathrm{true}$

$G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[3\,5\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,3\,5\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[2\,4\,7\,8\right]\right]\right)\right\}\right)$

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

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

$\mathrm{false}$

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

$\mathrm{false}$

$\mathrm{IsPrimitive}\left(G\,\left\{1\,3\,5\right\}\right)$

$\mathrm{true}$

$\mathrm{IsPrimitive}\left(G\,\left\{2\,4\,7\,8\right\}\right)$

$\mathrm{false}$

$\mathrm{BlockSystem}\left(G\,\left\{2\,4\,7\,8\right\}\right)$

$\left\{\left\{2\,7\right\}\,\left\{4\,8\right\}\right\}$

$\mathrm{BlockSystem}\left(G\,\left\{2\,4\,7\,8\right\}\,'\mathrm{containing}'=\left\{2\,4\right\}\right)$

$\left\{\left\{2\,4\,7\,8\right\}\right\}$

$G≔\mathrm{RubiksCubeGroup}\left(\right)$

$G≔⟨\left(6\,25\,43\,16\right)\left(7\,28\,42\,13\right)\left(8\,30\,41\,11\right)\left(17\,19\,24\,22\right)\left(18\,21\,23\,20\right)\,\left(1\,14\,48\,27\right)\left(2\,12\,47\,29\right)\left(3\,9\,46\,32\right)\left(33\,35\,40\,38\right)\left(34\,37\,39\,36\right)\,\left(1\,17\,41\,40\right)\left(4\,20\,44\,37\right)\left(6\,22\,46\,35\right)\left(9\,11\,16\,14\right)\left(10\,13\,15\,12\right)\,\left(3\,38\,43\,19\right)\left(5\,36\,45\,21\right)\left(8\,33\,48\,24\right)\left(25\,27\,32\,30\right)\left(26\,29\,31\,28\right)\,\left(1\,3\,8\,6\right)\left(2\,5\,7\,4\right)\left(9\,33\,25\,17\right)\left(10\,34\,26\,18\right)\left(11\,35\,27\,19\right)\,\left(14\,22\,30\,38\right)\left(15\,23\,31\,39\right)\left(16\,24\,32\,40\right)\left(41\,43\,48\,46\right)\left(42\,45\,47\,44\right)⟩$

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

$\mathrm{false}$

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

$\mathrm{false}$

$\mathrm{IsPrimitive}\left(G\,\mathrm{Elements}\left(\mathrm{Orbit}\left(1\,G\right)\right)\right)$

$\mathrm{false}$

$B≔\mathrm{MinimalBlockSystem}\left(G\,\mathrm{Elements}\left(\mathrm{Orbit}\left(1\,G\right)\right)\right)$

$B≔\left\{\left\{1\,9\,35\right\}\,\left\{3\,27\,33\right\}\,\left\{6\,11\,17\right\}\,\left\{8\,19\,25\right\}\,\left\{14\,40\,46\right\}\,\left\{16\,22\,41\right\}\,\left\{24\,30\,43\right\}\,\left\{32\,38\,48\right\}\right\}$

$S≔\mathrm{BlocksImage}\left(G\,B\right)$

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

$\mathrm{IsSymmetric}\left(S\right)$

$\mathrm{true}$

$\mathrm{IsPrimitive}\left(\mathrm{WreathProduct}\left(\mathrm{Symm}\left(23\right)\,\mathrm{Symm}\left(1000\right)\right)\right)$

$\mathrm{false}$

$G≔\mathrm{MathieuGroup}\left(24\right)$

$G≔M_{24}$

$\mathrm{BlockSystem}\left(G\,'\mathrm{containing}'=\left\{11\right\}\right)$

$\left\{\left\{1\right\}\,\left\{2\right\}\,\left\{3\right\}\,\left\{4\right\}\,\left\{5\right\}\,\left\{6\right\}\,\left\{7\right\}\,\left\{8\right\}\,\left\{9\right\}\,\left\{10\right\}\,\left\{11\right\}\,\left\{12\right\}\,\left\{13\right\}\,\left\{14\right\}\,\left\{15\right\}\,\left\{16\right\}\,\left\{17\right\}\,\left\{18\right\}\,\left\{19\right\}\,\left\{20\right\}\,\left\{21\right\}\,\left\{22\right\}\,\left\{23\right\}\,\left\{24\right\}\right\}$

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

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