38.9 Block Systems

computes a block system for the action. If seed is not given and the action is imprimitive, a minimal nontrivial block system will be found. If seed is given, a block system in which seed is the subset of one block is computed. The action must be transitive.

gap> g:=TransitiveGroup(8,3);
E(8) = 2[x]2[x]2
gap> Blocks(g,[1..8]);      
[ [ 1, 8 ], [ 2, 3 ], [ 4, 5 ], [ 6, 7 ] ]
gap> Blocks(g,[1..8],[1,4]);
[ [ 1, 4 ], [ 2, 7 ], [ 3, 6 ], [ 5, 8 ] ]

returns a block system that is maximal with respect to inclusion. maximal with respect to inclusion) for the action of G on Omega. If seed is given, a block system in which seed is the subset of one block is computed.

gap> MaximalBlocks(g,[1..8]);
[[1,2,3,8],[4,5,6,7]]

computes a list of block representatives for all minimal (i.e blocks are minimal with respect to inclusion) nontrivial block systems for the action.

gap> RepresentativesMinimalBlocks(g,[1..8]);
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], [ 1, 8 ] ]

computes a list of representatives of all block systems for a permutation group G acting transitively on the points moved by the group.

gap> AllBlocks(g);       
[ [ 1, 8 ], [ 1, 2, 3, 8 ], [ 1, 4, 5, 8 ], [ 1, 6, 7, 8 ], [ 1, 3 ], 
  [ 1, 3, 5, 7 ], [ 1, 3, 4, 6 ], [ 1, 5 ], [ 1, 2, 5, 6 ], [ 1, 2 ], 
  [ 1, 2, 4, 7 ], [ 1, 4 ], [ 1, 7 ], [ 1, 6 ] ]

The stabilizer of a block can be computed via the action OnSets (see OnSets):

gap> Stabilizer(g,[1,8],OnSets);
Group([ (1,8)(2,3)(4,5)(6,7) ])

If bs is a partition of omega, given as a set of sets, the stabilizer under the action OnSetsDisjointSets (see OnSetsDisjointSets) returns the largest subgroup which preserves bs as a block system.

gap> g:=Group((1,2,3,4,5,6,7,8),(1,2));;
gap> bs:=[[1,2,3,4],[5,6,7,8]];;
gap> Stabilizer(g,bs,OnSetsDisjointSets);
Group([ (5,6)(7,8), (5,8)(6,7), (1,5)(2,6)(3,7)(4,8), (1,2)(3,4), (1,4)(2,3), 
  (6,8,7), (2,3,4), (3,4), (7,8) ])

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