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# 3 Generic group classes

### Sections

1. Creating group classes
2. Properties of group classes
3. Additional properties of group classes
4. Attributes of group classes

In this chapter, we describe how group classes can be defined by assigning basic attributes and properties, in particular closure properties.

A class (see Set theoretical classes) is a group class if it consists of groups, and if it is closed under isomorphisms of groups. Thus if G and H are isomorphic groups, then G is in the group class grpclass if and only if H is. Groups belonging to the same group class may be regarded as sharing a group theoretical property (a property shared by isomorphic groups), and groups sharing a group theoretical property form a class of groups. It not empty, group classes are genuinely infinite objects, so GAP sets can never form group classes. Some authors require every group class to contain the trivial groups. Here we do not make this assumption; in particular every empty class is a group class.

The following sections describe how to create group classes and declare some of their basic properties.

Note that, for common types of group classes, there are additional functions available to accomplish this task; see the following Chapters Schunck classes and formations and Fitting classes and Fitting sets. There are also a number of pre-defined group classes; see Chapter Examples of group classes.

## 3.1 Creating group classes

Group classes can either be defined by a function deciding membership, or alternatively by a (finite) list of groups containing at least one representative of each isomorphism type of groups contained in the class.

- `GroupClass(`rec`) O`
- `GroupClass(`func`) O`
- `GroupClass(`group-list`) O`
- `GroupClass(`group-list`,` iso-func`) O`

  The function `GroupClass` returns a new group class class, specified by its argument(s).

  In the first form, rec must be a record which has a component `\in`, and may have further components `name`, and `char`. `\in` must be a function having one argument. When called with a group G as its argument, it must return `true` or `false`, depending upon whether G is in class or not. It is the user's responsibility to ensure that the function supplied returns the same value when called with isomorphic groups. If rec has components `name` or `char`, their values are stored in the attributes `Name` (see Name) and `Characteristic` (see Characteristic) of class.

  `GroupClass(`func`)` is a shorthand for `GroupClass(rec(\in := func))`.

  In the other cases, `GroupClass` returns the group class consisting of the isomorphism classes of the groups in the list group-list. If iso-func is given, iso-func is used to check whether a given group G is isomorphic with one of the groups in the defining list. iso-func must have two arguments, and must return true if two groups, one of which is in group-list, passed as arguments are isomorphic, and false otherwise. If iso-func is not given, the GAP function `IsomorphismGroups` is used for the isomorphism test. Note that even for relatively small groups, `IsomorphismGroups` tends to be very slow.

      gap> GroupClass(IsNilpotent);
      GroupClass (in:=<Operation "IsNilpotent">)
      gap> GroupClass([CyclicGroup(2), CyclicGroup(3)]);
      GroupClass ([ <pc group of size 2 with 1 generators>,
        <pc group of size 3 with 1 generators> ])
      gap> AbelianIsomorphismTest := function (A,B)
      >     if IsAbelian (A) then
      >         if IsAbelian (B) then
      >             return AbelianInvariants (A) = AbelianInvariants (B);
      >         else
      >             return false;
      >         fi;
      >     elif IsAbelian (B) then
      >         return false;
      >     else # this will not happen if called from GroupClass
      >         Error ("At least one of the groups <A> and <B> must be abelian");
      >     fi;
      > end;
      function( A, B ) ... end
      gap> cl := GroupClass ([AbelianGroup ([2,2]), AbelianGroup ([3,5])],
      > AbelianIsomorphismTest);
      GroupClass ([ <pc group of size 4 with 2 generators>,
        <pc group of size 15 with 2 generators> ], function( A, B ) ... end)
      gap> Group ((1,2), (3,4)) in cl;
      true

- `Intersection(`list`) F`
- `Intersection(`C_1`,` C_2`, ...,` C_n`) F`

  The intersection of a list list of group classes resp. of the group classes C_1, C_2, ..., C_n is again a group class. The intersection automatically has those closure properties (see Properties of group classes) which all of the intersected classes have.

  ## 3.2 Properties of group classes

  Since nonempty group classes are infinite, CRISP cannot, in general, decide whether a group class has a certain property. Therefore the user is required to set the appropriate properties and attributes. See Sections attributes and properties in the GAP reference manual. To facilitate this task, there are special functions available to create common types of group classes such as formations (see Creating formations), Fitting classes (see Creating Fitting classes), and Schunck classes (see Creating Schunck classes).

  However, CRISP knows about the implications between the closure properties listed below; for instance it knows that a group class which has `IsResiduallyClosed` also has `IsDirectProductClosed`, and that a class having `IsSchunckClass` also has `IsDirectProductClosed` and `IsSaturated`. Moreover, the intersection of group classes all having one of the closure properties in common also has that closure property.

  The following basic properties are defined for group classes.

- `IsGroupClass(`grpclass`) P`

  A generic class (see Chapter set theoretical classes) is considered a group class if it has the property `IsGroupClass`. There is no way for CRISP to know that a given class defined by a membership function is a group class, i. e., consists of groups and is closed under group isomorphisms.

- `ContainsTrivialGroup(`grpclass`) P`

  This property, if bound, indicates whether grpclass contains the trivial group or not.

- `IsSubgroupClosed(`grpclass`) P`

  if true, then for every G in grpclass, the subgroups of G likewise belong to grpclass.

- `IsNormalSubgroupClosed(`grpclass`) P`

  if true, then for every G in grpclass, the (sub)normal subgroups of G likewise belong to grpclass.

- `IsQuotientClosed(`grpclass`) P`

  if true, then for every G in grpclass, the factor groups of G likewise belong to grpclass.

- `IsResiduallyClosed(`grpclass`) P`

  if true and G is a group such that G/N1 and G/N2 belong to grpclass for two normal subgroups N1 and N2 of G which intersect trivially, then G belongs to grpclass.

- `IsNormalProductClosed(`grpclass`) P`

  if true and G is a group which is generated by subnormal subgroups in grpclass, then G belongs to grpclass.

- `IsDirectProductClosed(`grpclass`) P`

  if true and the group G is the direct product of N1 and N2 belonging to grpclass, then G likewise belongs to grpclass.

- `IsSchunckClass(`grpclass`) P`

  if true, then G belongs to grpclass if and only if its primitive factor groups lie in grpclass. A (finite) group is primitive if it has a faithful primitive permutation representation, or equivalently, if it has a maximal subgroup with trivial core. A Schunck class contains every trivial group.

- `IsSaturated(`grpclass`) P`

  if true, G belongs to X whenever G/`FrattiniSubgroup(`G`)` belongs to X.

  ## 3.3 Additional properties of group classes

  Note that the following ``properties'' are not properties but only filters in the GAP sense (cf. Properties and filters in the GAP reference manual).

- `HasIsFittingClass(`obj`) F`

  is true if obj knows if it is a Fitting class, that is, if it lies in the filters `HasIsGroupClass`, `HasContainsTrivialGroup`, `HasIsNormalSubgroupClosed` and `HasIsNormalProductClosed`.

- `IsFittingClass(`obj`) F`

  is true if obj is a Fitting class, that is, if it has the properties `IsGroupClass`, `ContainsTrivialGroup`, `IsNormalSubgroupClosed` and `IsNormalProductClosed`.

- `SetIsFittingClass(`group class`,` bool`) F`

  If bool is `true`, this fake setter function sets the properties `IsNormalSubgroupClosed` and `IsNormalProductClosed` of group class to true. It is the user's responsibility to ensure that group class is indeed a Fitting class.

      gap> nilp := GroupClass (IsNilpotent);
      GroupClass (in:=<Operation "IsNilpotent">)
      gap> SetIsFittingClass (nilp, true);
      gap> nilp;
      FittingClass (in:=<Operation "IsNilpotent">)

- `HasIsOrdinaryFormation(`obj`) F`

  `HasIsGroupClass`, `HasContainsTrivialGroup`, `HasIsQuotientClosed` and `HasIsResiduallyClosed`.

- `IsOrdinaryFormation(`obj`) F`

  `IsGroupClass`, `ContainsTrivialGroup`, `IsQuotientClosed` and `IsResiduallyClosed`.

- `SetIsOrdinaryFormation(`class`,` bool`) F`

  `ContainsTrivialGroup`, and `IsResiduallyClosed` of class, making it a formation.

- `HasIsSaturatedFormation(`obj`) F`

  returns true if obj knows if it is a saturated formation, that is, if it lies in the filters `HasIsOrdinaryFormation` and `HasIsSaturated`.

- `IsSaturatedFormation(`obj`) F`

  returns true if obj is a saturated formation, that is, if it has the properties `IsOrdinaryFormation` and `IsSaturated`

- `SetIsSaturatedFormation(`class`,` bool`) F`

  If bool is `true`, this sets the attributes `IsQuotientClosed`, `ContainsTrivialGroup`, and `IsResiduallyClosed` and `IsSaturated` of class, making it a saturated formation.

- `HasIsFittingFormation(`obj`) F`

  returns true if obj knows whether it is a Fitting formation, that is, if it lies in the filters `HasIsOrdinaryFormation` and `HasIsFittingClass` (see HasIsOrdinaryFormation and HasIsFittingClass).

- `IsFittingFormation(`obj`) F`

  returns true if obj is both a formation and a Fitting class.

- `SetIsFittingFormation(`class`,` bool`) F`

  If bool is `true`, this function sets the attributes of class to indicate that it is a Fitting formation.

- `HasIsSaturatedFittingFormation(`obj`) F`

  returns true if obj knows whether it is a saturated Fitting formation, that is, if it lies in the filters `HasIsSaturatedFormation` and `HasIsFittingClass` (see HasIsSaturatedFormation and HasIsFittingClass).

- `IsSaturatedFittingFormation(`obj`) F`

  returns true if obj is both a saturated formation and a Fitting class, that is, if it lies in the filters `IsSaturatedFormation` and `IsFittingClass` (see IsSaturatedFormation and IsFittingClass).

- `SetIsSaturatedFittingFormation(`class`,` bool`) F`

  If bool is `true`, this sets the attributes of class to indicate that it is a saturated Fitting formation.

  ## 3.4 Attributes of group classes

  In addition to the attribute `MemberFunction` which has the same meaning as for generic classes, a group class may have the following attribute.

- `Characteristic(`grpclass`) A`

  This attribute, if present, stores a class containing all primes p such that grpclass contains a cyclic group of order p. There is a pre-defined class `AllPrimes` which should be assigned to `Characteristic` if grpclass contains a cyclic group of order p for every prime p.

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