[FORMAT] 2 Formations in GAP

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# 2 Formations in GAP

A formation is a class F of groups closed under taking epimorphic images and subdirect products. Closure under subdirect products is equivalent to the property that each finite group G has a unique smallest normal subgroup GF with factor group G / GF in F. The subgroup GF is called the F-residual subgroup of G. Thus, for example, the derived subgroup of G is its residual for the formation of abelian groups, and the residual for the formation of nilpotent groups is the last term of the descending central series.

In FORMAT a formation is described by a function that computes GF for each (finite solvable) group G, and from that perspective F consists of the groups G for which GF is trivial. To define a formation that is not one of the standard examples provided (see below), one must give GAP an identifier for the formation and also some method for computing residual subgroups.

Some of the most interesting formations can also be described by ``local definition.'' For each prime p let F(p) be a formation or the empty class, and let F be the class of all finite solvable groups G such that for each prime p and each p-chief factor H/K of G the group of automorphisms that G induces on H/K by conjugation belongs to F(p). Then F is a formation, with local definition the set of F(p)s. The set of primes p for which F(p) is not empty is called the support of F. A p-chief factor is F-central in case G induces an F(p)-group on it or, equivalently, in case GF(p) centralizes it. It is possible to define a formation in FORMAT by giving such a local definition. Indeed one can define a kind of generalized formation by giving what is called a normal subgroup function or screen, which specifies arbitrary normal subgroups, not necessarily of form GF(p), to test ``centrality.'' Section Other Applications describes one such usage of general screens. Most applications of formation theory to solvable groups require local definition, as do the GAP functions for computing F-normalizers and F-covering subgroups.

- `Formation(` rec`) O`
- `Formation(` str`[,` primes`] ) O`

  The definition of a formation in FORMAT begins with the creation of a record `rec`, which must contain a `name` component and at least one of the components `fResidual` or `fScreen`. The component `name` is a string, `fResidual` is a function that computes a normal subgroup of each group, and `fScreen` is a function of two variables, a group and a prime, that returns a normal subgroup of the input group.

  In the second form the function `Formation` can be used to obtain a formation from the supplied library of formations. The formations provided are:

  `Formation( "Nilpotent" )`
    The formation of nilpotent groups,

  `Formation( "Supersolvable" )`
    The formation of supersolvable groups,

  `Formation( "Abelian" )`
    The formation of abelian groups,

  `Formation( "ElementaryAbelianProduct" )`
    The formation of direct products of elementary abelian groups,

  `Formation( "PNilpotent", prime )`
    The formation of p-nilpotent groups for p = `prime`,

  `Formation( "PiGroups", primes )`
    The formation of π-groups for π = the set `primes`,

  `Formation( "PLengthOne", prime )`
    The formation of groups of p-length 1 for p = `prime`.

- `IsFormation(` F`) C`
- `NameOfFormation(` F`) A`
- `ResidualFunctionOfFormation(` F`) A`

  `IsFormation` returns `true` if and only if F is a GAP formation. `NameOfFormation` returns the name of a formation and `ResidualFunctionOfFormation` returns the residual function of a formation.

- `ScreenOfFormation(` F`) A`

  If F is locally defined by some screen of F(p)s, then `HasScreenOfFormation(` F`)` is `true`, `ScreenOfFormation(` F`)` is a function of two variables, group and prime, and `ScreenOfFormation(` F`)(` G`,` p`)` returns GF(p) if p is in the support of F and gives the empty list otherwise.

- `SupportOfFormation(` F`) A`

  The attribute `SupportOfFormation` is optional. It may be bound by `SetSupportOfFormation`. If `SupportOfFormation` is not bound, then the support of the formation is taken to be the set of all primes. In case the support of F is a finite set of primes, then `SupportOfFormation(` F`)` is a list of those primes, and `HasSupportOfFormation(` F`)` returns true. In case the support of F is an infinite set but not the set of all primes, then the user will need to make sure, perhaps with `ChangedSupport` or `SetSupportOfFormation`, that all primes dividing the orders of relevant groups are considered.

- `ChangedSupport(` F`,` primes`) O`

  This function may be used to change the support of a formation. Let F be a formation and primes a list of primes. Then `ChangedSupport` returns a formation with a new name whose support is the intersection of the support of F and primes.

- `IsIntegrated(` F`) P`

  The local definition is called integrated in case F(p) is contained in F for each prime p. The optional property `IsIntegrated` makes sense only if `HasScreenOfFormation(` F`)` is `true`. Notice that some of the functions described below will require that all of the attributes `HasScreenOfFormation(` F`)`, `HasIsIntegrated(` F`)` and `IsIntegrated(` F`)` are `true`. If unbound, this property can be bound with `SetIsIntegrated`, but it is up to the user to determine whether such a setting is appropriate. Section Formation Examples contains an example of such usage.

- `Integrated(` F`) O`

  A local definition of a formation may always be replaced by an integrated one without changing the formation itself, though the meaning of F-central may change. Let F be a locally defined GAP formation with name ``name``. If F is already integrated, then `Integrated(` F`)` yields F itself. Otherwise, it yields a formation ``name`Int` that is abstractly the same as F but has integrated local definition.

- ``F1`=` F2``
- ``F1`<` F2``

  Two GAP formations F1 and F2 are considered to be equal in case they have the same name. The natural ordering on strings gives an ordering on formations. This ordering is useful for organizing key-dependent lists but has no mathematical significance.

- `Intersection(` F1`,` F2`) O`

  The intersection of two GAP formations F1 and F2 is again a formation. `Intersection` produces the new formation `(`name1`And`name2`)`, which has attribute `ResidualFunctionOfFormation` if either F1 or F2 does, has `FScreen` whenever both formations have `FScreen`, and is integrated if both are.

- `ProductOfFormations(` F1`,` F2`) O`

  The product of two formations F1 and F2 is the formation F such that a finite group G is a member of F if and only if GF2 is in F1. (Notice that the product of F1 by F2 is not necessarily equal to the product of F2 by F1, and unless F1 is normal subgroup-closed the product need not contain all extensions of a group in F1 by a group in F2.) The function `ProductOfFormations(` F1`,` F2`)` yields the product `(`name1`By`name2`)` of the two formations. The product has the attribute `ResidualFunctionOfFormation` and has the attribute `ScreenOfFormation` whenever both F1 and F2 have this entry or whenever both `HasScreenOfFormation(` F2`)` and `not HasSupportOfFormation(` F1`)` are `true`. In these cases the property `IsIntegrated` will be inherited if possible.

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