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6 Formation Examples

The following is a GAP session that illustrates the various functions in the package. We have chosen to work with the symmetric group S4 and the special linear group SL(2,3) as examples, because it is easy to print and read the results of computations for these groups, and the answers can be checked by inspection. However, both S4 and SL(2,3) are extremely small examples for the algorithms in FORMAT. In EW we describe effective application of the algorithms to groups of composition length as much as 61, for which the computations take a few seconds to complete. The file grp contains some of these groups and other groups readable as GAP input. 

gap> LoadPackage("format");;
A primitive banner appears.

First we define S4 as a permutation group and compute some subgroups of it. 

gap> G := SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> SystemNormalizer(G);  CarterSubgroup(G);
Group([ (3,4) ])
Group([ (3,4), (1,3)(2,4), (1,2)(3,4) ])
Now we take the formation of supersolvable groups from the examples and look at it. 
gap> sup := Formation("Supersolvable");
formation of Supersolvable groups
gap> KnownAttributesOfObject(sup); KnownPropertiesOfObject(sup);
[ "NameOfFormation", "ScreenOfFormation" ]
[ "IsIntegrated" ]

We can look at the screen for sup. 

gap> ScreenOfFormation(sup);
<Operation "AbelianExponentResidual">
gap> ScreenOfFormation(sup)(G,2); ScreenOfFormation(sup)(G,3);
Group([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ])
Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ])
We get the residuals for G of the formations of abelian groups of exponent 1 ( = 2−1) and of exponent 2 (=3−1).

Notice that sup does not yet have a residual function. Let's compute some subgroups of G corresponding to sup. 

gap> ResidualWrtFormation(G, sup);
Group([ (1,2)(3,4), (1,4)(2,3) ])
gap> KnownAttributesOfObject(sup);
[ "NameOfFormation", "ScreenOfFormation", "ResidualFunctionOfFormation" ]
The residual function for sup was required and created. 
gap> FNormalizerWrtFormation(G, sup);
Group([ (3,4), (2,4,3) ])
gap> CoveringSubgroupWrtFormation(G, sup);
Group([ (3,4), (2,4,3) ])
gap> KnownAttributesOfObject(G);
[ "Size", "OneImmutable", "SmallestMovedPoint", "NrMovedPoints", 
  "MovedPoints", "GeneratorsOfMagmaWithInverses", "TrivialSubmagmaWithOne", 
  "MultiplicativeNeutralElement", "DerivedSubgroup", "IsomorphismPcGroup", 
  "IsomorphismSpecialPcGroup", "PcgsElementaryAbelianSeries", "Pcgs", 
  "GeneralizedPcgs", "StabChainOptions", "ComputedResidualWrtFormations", 
  "ComputedAbelianExponentResiduals", "ComputedFNormalizerWrtFormations", 
  "ComputedCoveringSubgroup1s", "ComputedCoveringSubgroup2s", 
  "SystemNormalizer", "CarterSubgroup" ]
The AbelianExponentResiduals were computed in connection with the local definition of sup. (AbelianExponentResidual(G, n) returns the smallest normal subgroup of G whose factor group is abelian of exponent dividing n-1.) Here are some of the other records. 
gap> ComputedResidualWrtFormations(G);
[ formation of Supersolvable groups, Group([ (1,2)(3,4), (1,4)(2,3) ]) ]
gap> ComputedFNormalizerWrtFormations(G);
[ formation of Nilpotent groups, Group([ (3,4) ]), 
  formation of Supersolvable groups, Group([ (3,4), (2,4,3) ]) ]
gap> ComputedCoveringSubgroup2s(G);
[  ]
gap> ComputedCoveringSubgroup1s(G);
[ formation of Nilpotent groups, Group([ (3,4), (1,3)(2,4), (1,2)(3,4) ]), 
  formation of Supersolvable groups, Group([ (3,4), (2,4,3) ]) ]
The call by CoveringSubgroupWrtFormation was to CoveringSubgroup1, not CoveringSubgroup2.

We could also have started with a pc group or a nice enough matrix group. 

gap> s4 := SmallGroup(IdGroup(G));
<pc group of size 24 with 4 generators>
This is S4 again. The answers just look different now. 
gap> SystemNormalizer(s4); CarterSubgroup(s4);
Group([ f1 ])
Group([ f1, f4, f3*f4 ])
Similarly, we have SL(2,3) and an isomorphic pc group. 
gap> sl := SpecialLinearGroup(2,3);
SL(2,3)
gap> h := SmallGroup(IdGroup(sl));
<pc group of size 24 with 4 generators>
We get the following subgroups. 
gap> CarterSubgroup(sl); Size(last);
<group of 2x2 matrices in characteristic 3>
6
gap> SystemNormalizer(h); CarterSubgroup(h);
Group([ f1, f4 ])
Group([ f1, f4 ])

Now let's make new formations from old. 

gap> ab := Formation("Abelian");
formation of Abelian groups
gap> KnownPropertiesOfObject(ab); KnownAttributesOfObject(ab);
[  ]
[ "NameOfFormation", "ResidualFunctionOfFormation" ]
gap> nil2 := Formation("PNilpotent",2);
formation of 2Nilpotent groups
gap> KnownPropertiesOfObject(nil2); KnownAttributesOfObject(nil2);
[ "IsIntegrated" ]
[ "NameOfFormation", "ScreenOfFormation", "ResidualFunctionOfFormation" ]
Compute the product and check some attributes. 
gap> form := ProductOfFormations(ab, nil2);
formation of (AbelianBy2Nilpotent) groups
gap> KnownAttributesOfObject(form);
[ "NameOfFormation", "ResidualFunctionOfFormation" ]
Now the product in the other order, which is locally defined. 
gap> form2 := ProductOfFormations(nil2, ab);
formation of (2NilpotentByAbelian) groups
gap> KnownAttributesOfObject(form2);
[ "NameOfFormation", "ScreenOfFormation", "ResidualFunctionOfFormation" ]
We check the results on G, which is still S4. 
gap> ResidualWrtFormation(G, form);  ResidualWrtFormation(G, form2);
Group(())
Group([ (1,3)(2,4), (1,2)(3,4) ])
gap> KnownPropertiesOfObject(form2);
[  ]
Although form2 is not integrated, we can make an integrated formation that differs from form2 only in its local definition, i.e., whose residual subgroups are the same as those for form2. 
gap> Integrated(form2);
formation of (2NilpotentByAbelian)Int groups
FNormalizerWrtFormation and CoveringSubgroupWrtFormation both require integrated formations, so they silently replace form2 by this last formation without, however, changing form2. 
gap> FNormalizerWrtFormation(G, form2); CoveringSubgroupWrtFormation(G, form2);
Group([ (3,4), (2,4,3) ])
Group([ (3,4), (2,4,3) ])
gap> KnownPropertiesOfObject(form2);
[  ]
gap> ComputedCoveringSubgroup1s(G);
[ formation of (2NilpotentByAbelian)Int groups, Group([ (3,4), (2,4,3) ]), 
  formation of Nilpotent groups, Group([ (3,4), (1,3)(2,4), (1,2)(3,4) ]), 
  formation of Supersolvable groups, Group([ (3,4), (2,4,3) ]) ]
gap> ComputedResidualWrtFormations(G);
[ formation of (2NilpotentByAbelian) groups, 
  Group([ (1,4)(2,3), (1,2)(3,4) ]), 
  formation of (AbelianBy2Nilpotent) groups, Group(()), 
  formation of 2Nilpotent groups, Group([ (1,2)(3,4), (1,3)(2,4) ]), 
  formation of Abelian groups, Group([ (2,4,3), (1,4)(2,3), (1,3)(2,4) ]), 
  formation of Supersolvable groups, Group([ (1,2)(3,4), (1,4)(2,3) ]) ]
Lots of work has been going on behind the scenes.

Before we compute an intersection, we construct yet another formation. 

gap> pig := Formation("PiGroups", [2,5]);
formation of (2,5)-Group groups with support [ 2, 5 ]
gap> form := Intersection(pig, nil2);
formation of ((2,5)-GroupAnd2Nilpotent) groups with support [ 2, 5 ]
gap> KnownAttributesOfObject(form);
[ "NameOfFormation", "ScreenOfFormation", "SupportOfFormation", 
  "ResidualFunctionOfFormation" ]
Let's cut down the support of nil2 to {2,5}. 
gap> form3 := ChangedSupport(nil2, [2,5]);
formation of Changed2Nilpotent[ 2, 5 ] groups
gap> SupportOfFormation(form3);
[ 2, 5 ]
gap> form = form3;
false
Although the formations defined by form and form3 are abstractly identical, GAP has no way to know this fact, and so distinguishes them.

We can mix the various operations, too. 

gap> ProductOfFormations(Intersection(pig, nil2), sup);
formation of (((2,5)-GroupAnd2Nilpotent)BySupersolvable) groups
gap> Intersection(pig, ProductOfFormations(nil2, sup));
formation of ((2,5)-GroupAnd(2NilpotentBySupersolvable)) groups with support 
[ 2, 5 ]

Now let's define our own formation. 

gap> preform := rec( name := "MyOwn", 
>  fScreen := function( G, p)
>  return DerivedSubgroup( G );
>  end);
rec( fScreen := function( G, p ) ... end, name := "MyOwn" )
gap> form := Formation(preform);
formation of MyOwn groups
gap> KnownAttributesOfObject(form); KnownPropertiesOfObject(form);
[ "NameOfFormation", "ScreenOfFormation" ]
[  ]
In fact, the definition is integrated. Let's tell GAP so and compute some related subgroups. 
gap> SetIsIntegrated(form, true);
gap> ResidualWrtFormation(G, form);
Group([ (1,4)(2,3), (1,2)(3,4) ])
gap> FNormalizerWrtFormation(G, form);
Group([ (3,4), (2,4,3) ])
gap> CoveringSubgroup1(G, form);
Group([ (3,4), (2,4,3) ])
These answers are consistent with the fact that MyOwn is really just the formation of abelian by nilpotent groups.

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FORMAT manual
January 2024