38.8 Tests for Actions

  • IsTransitive( G, Omega[, gens, acts][, act] ) O
  • IsTransitive( xset ) P

    An action is transitive if the whole domain forms one orbit.

  • Transitivity( G, Omega[, gens, acts][, act] ) O
  • Transitivity( xset ) A

    An action is k-transitive if every k-tuple of points can be mapped simultaneously to every other k-tuple. 

    gap> g:=Group((1,3,2),(2,4,3));;
gap> IsTransitive(g,[1..5]);
false
gap> Transitivity(g,[1..4]);
2
    Note: For permutation groups, the syntax IsTransitive(g) is also permitted and tests whether the group is transitive on the points moved by it, that is the group á(2,3,4),(2,3)ñ is transitive (on 3 points).

  • RankAction( G, Omega[, gens, acts][, act] ) O
  • RankAction( xset ) A

    The rank of a transitive action is the number of orbits of the point stabilizer. 

    gap> RankAction(g,Combinations([1..4],2),OnSets);
4
  • IsSemiRegular( G, Omega[, gens, acts][, act] ) O
  • IsSemiRegular( xset ) P

    An action is semiregular is the stabilizer of each point is the identity.

  • IsRegular( G, Omega[, gens, acts][, act] ) O
  • IsRegular( xset ) P

    An action is regular if it is semiregular (see IsSemiRegular) and transitive. In this case every point pnt of Omega defines a one to one correspondence between G and Omega. 

    gap> IsSemiRegular(g,Arrangements([1..4],3),OnTuples);
true
gap> IsRegular(g,Arrangements([1..4],3),OnTuples);    
false
  • Earns( G, Omega[, gens, acts][, act] ) O
  • Earns( xset ) A

    returns a list of the elementary abelian regular (when acting on Omega) normal subgroups of G.

  • IsPrimitive( G, Omega[, gens, acts][, act] ) O
  • IsPrimitive( xset ) P

    An action is primitive if it is transitive and no nontrivial block systems are permissible. See Block Systems. 

    gap> IsPrimitive(g,Orbit(g,(1,2)(3,4))); 
true

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    GAP 4 manual
    February 2000