 BlocksImage - Maple Help

GroupTheory

 IsPrimitive
 determine whether a permutation group is primitive
 BlockSystem
 return a non-trivial block system for a transitive group if one exists
 MinimalBlockSystem
 return a minimal non-trivial block system for a transitive group if one exists
 BlocksImage
 return a permutation group equivalent to the action of a transitive group on a block system Calling Sequence IsPrimitive( G, domain ) BlockSystem( G, domain ) BlockSystem( G, domain, containing = S ) MinimalBlockSystem( G, domain ) BlocksImage( G, B ) Parameters

 G - : PermutationGroup : a permutation group S - : set(posint) : a subset of the support of G B - : set(set(posint)) : a block system for G domain - : set(posint) : (optional) a G-invariant subset of the support of G 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.
 • 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. Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{PermutationGroup}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,2,3\right],\left[4,5\right]\right]\right)\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (1)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{false}}$ (2)
 > $\mathrm{IsPrimitive}\left(\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1,2,4\right]\right]\right),\mathrm{Perm}\left(\left[\left[4,5,7\right]\right]\right)\right\}\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsPrimitive}\left(\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsPrimitive}\left(\mathrm{PGU}\left(3,3\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsPrimitive}\left(\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{IsPrimitive}\left(\mathrm{DihedralGroup}\left(5\right)\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsPrimitive}\left(\mathrm{SuzukiGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (8)

Use the optional domain argument to restrict the action.

 > $G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1,3,5\right]\right]\right),\mathrm{Perm}\left(\left[\left[3,5\right]\right]\right),\mathrm{Perm}\left(\left[\left[2,4,7,8\right]\right]\right)\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{3}{,}{5}\right){,}\left({2}{,}{4}{,}{7}{,}{8}\right){,}\left({3}{,}{5}\right)⟩$ (9)

This group is not primitive on its support.

 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{false}}$ (10)

It does not even act transitively.

 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{false}}$ (11)

Restricting to the $G$-stable subset $\left\{1,3,5\right\}$, we obtain a primitive action.

 > $\mathrm{IsPrimitive}\left(G,\left\{1,3,5\right\}\right)$
 ${\mathrm{true}}$ (12)

However, the action on the $G$-stable subset $\left\{2,4,7,8\right\}$ is not primitive.

 > $\mathrm{IsPrimitive}\left(G,\left\{2,4,7,8\right\}\right)$
 ${\mathrm{false}}$ (13)

A block system demonstrating that $G$ is not primitive can be obtained as follows.

 > $\mathrm{BlockSystem}\left(G,\left\{2,4,7,8\right\}\right)$
 $\left\{\left\{{2}{,}{7}\right\}{,}\left\{{4}{,}{8}\right\}\right\}$ (14)

However, there is no block system in which $\left\{2,4\right\}$ is contained in a non-trivial block.

 > $\mathrm{BlockSystem}\left(G,\left\{2,4,7,8\right\},'\mathrm{containing}'=\left\{2,4\right\}\right)$
 $\left\{\left\{{2}{,}{4}{,}{7}{,}{8}\right\}\right\}$ (15)

Consider the group of the Rubik's cube.

 > $G≔\mathrm{RubiksCubeGroup}\left(\right)$
 ${G}{≔}{\mathrm{< a permutation group on 48 letters with 6 generators >}}$ (16)

This group is not primitive

 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{false}}$ (17)

That is because it does not even act transitively.

 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{false}}$ (18)

However, restricting to one of its (two) orbits, upon which it of course acts transitively, it is still not primitive.

 > $\mathrm{IsPrimitive}\left(G,\mathrm{Elements}\left(\mathrm{Orbit}\left(1,G\right)\right)\right)$
 ${\mathrm{false}}$ (19)
 > $B≔\mathrm{MinimalBlockSystem}\left(G,\mathrm{Elements}\left(\mathrm{Orbit}\left(1,G\right)\right)\right)$
 ${B}{≔}\left\{\left\{{1}{,}{9}{,}{35}\right\}{,}\left\{{3}{,}{27}{,}{33}\right\}{,}\left\{{6}{,}{11}{,}{17}\right\}{,}\left\{{8}{,}{19}{,}{25}\right\}{,}\left\{{14}{,}{40}{,}{46}\right\}{,}\left\{{16}{,}{22}{,}{41}\right\}{,}\left\{{24}{,}{30}{,}{43}\right\}{,}\left\{{32}{,}{38}{,}{48}\right\}\right\}$ (20)

Because $B$ is a minimal block system (consisting of maximal blocks), the blocks image of $G$ on $B$ is primitive.

 > $S≔\mathrm{BlocksImage}\left(G,B\right)$
 ${S}{≔}⟨\left({3}{,}{4}{,}{7}{,}{6}\right){,}\left({1}{,}{5}{,}{8}{,}{2}\right){,}\left({1}{,}{3}{,}{6}{,}{5}\right)⟩$ (21)
 > $\mathrm{IsSymmetric}\left(S\right)$
 ${\mathrm{true}}$ (22)
 > $\mathrm{IsPrimitive}\left(\mathrm{WreathProduct}\left(\mathrm{Symm}\left(23\right),\mathrm{Symm}\left(1000\right)\right)\right)$
 ${\mathrm{false}}$ (23)
 > $G≔\mathrm{MathieuGroup}\left(24\right)$
 ${G}{≔}{{M}}_{{24}}$ (24)
 > $\mathrm{BlockSystem}\left(G,'\mathrm{containing}'=\left\{11\right\}\right)$
 $\left\{\left\{{1}\right\}{,}\left\{{2}\right\}{,}\left\{{3}\right\}{,}\left\{{4}\right\}{,}\left\{{5}\right\}{,}\left\{{6}\right\}{,}\left\{{7}\right\}{,}\left\{{8}\right\}{,}\left\{{9}\right\}{,}\left\{{10}\right\}{,}\left\{{11}\right\}{,}\left\{{12}\right\}{,}\left\{{13}\right\}{,}\left\{{14}\right\}{,}\left\{{15}\right\}{,}\left\{{16}\right\}{,}\left\{{17}\right\}{,}\left\{{18}\right\}{,}\left\{{19}\right\}{,}\left\{{20}\right\}{,}\left\{{21}\right\}{,}\left\{{22}\right\}{,}\left\{{23}\right\}{,}\left\{{24}\right\}\right\}$ (25) Compatibility

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