36.18 Sets of Subgroups

generates the conjugacy class of subgroups of G with representative U. This class is an external set, so functions such as Representative, (which returns U), ActingDomain (which returns G), StabilizerOfExternalSet (which returns the normalizer of U), and AsList work for it. It is possible to use the [] list access to select elements of the class. Because of potential other methods installed, the AsList command may give a different arrangement of the class elements!

gap> g:=Group((1,2,3,4),(1,2));;IsNaturalSymmetricGroup(g);;
gap> cl:=ConjugacyClassSubgroups(g,Subgroup(g,[(1,2)]));
Group( [ (1,2) ] )^G
gap> Size(cl);
6
gap> cl[4];
Group( [ (2,4) ] )

Is the representation GAP uses for conjugacy classes of subgroups. It can be used to check whether an object is a class of subgroups. The second representation IsConjugacyClassSubgroupsByStabilizerRep in addition is an external orbit by stabilizer and will compute its elements via a transversal of the stabilizer.

This attribute returns a list of conjugacy classes of subgroups of the group G. It also is applicable for lattices of subgroups (see LatticeSubgroups). The order in which the groups are listed depends on the method chosen by GAP.

gap> ConjugacyClassesSubgroups(g); 
[Group(())^G,Group([(1,4)(2,3)])^G,Group([(1,4)(2,3),(1,3)(2,4)])^G,
["omitted"],
Group([(1,2)(3,4),(1,2)])^G,Group([(1,2)(3,4),(1,3,2,4)])^G]

returns the conjugacy classes of maximal subgroups of G. Representatives of the classes can be computed directly by MaximalSubgroupClassReps (see MaximalSubgroupClassReps).

gap> ConjugacyClassesMaximalSubgroups(g);
[Group([(2,4,3),(1,4)(2,3),(1,2)(3,4)])^G,
 Group([(3,4),(1,4)(2,3),(1,2)(3,4)])^G, Group([(3,4),(2,4,3)])^G]

returns a list of conjugacy representatives of the maximal subgroups of G.

gap> MaximalSubgroupClassReps(g);   
[Group([(2,4,3),(1,4)(2,3),(1,2)(3,4)]),Group([(3,4),(1,4)(2,3),(1,2)(3,4)]),
Group([(3,4),(2,4,3)])]

returns a list of all maximal subgroups of G. This may take up much space, therefore the command should be avoided if possible. See ConjugacyClassesMaximalSubgroups.

gap> MaximalSubgroups(Group((1,2,3),(1,2)));
[Group([(1,2,3)]),Group([(2,3)]),Group([(1,3)]),Group([(1,2)])]

returns a list of all normal subgroups of G.

gap> g:=SymmetricGroup(4);;NormalSubgroups(g);
[ Group(()), Group([ (1,4)(2,3), (1,2)(3,4) ]), 
  Group([ (2,4,3), (1,4)(2,3), (1,2)(3,4) ]), Sym( [ 1 .. 4 ] ) ]

is a list containing those proper normal subgroups of the group G that are maximal among the proper normal subgroups.

gap> MaximalNormalSubgroups( g );
[ Group([ (2,4,3), (1,4)(2,3), (1,2)(3,4) ]) ]

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