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<html><head><title>[FORMAT] 2 Formations in GAP</title></head>
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<h1>2 Formations in GAP</h1>

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 <a href="CHAP007.htm">Other Applications</a> 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.

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<li><code>Formation( </code><var>rec</var><code> ) O</code>
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<li><code>Formation( </code><var>str</var><code> [, </code><var>primes</var><code> ] ) O</code>

The definition of a formation in FORMAT begins with the creation of a
record <code>rec</code>, which must contain a <code>name</code> component and at least one of
the components <code>fResidual</code> or <code>fScreen</code>. The component <code>name</code> is a string,
<code>fResidual</code> is a function that computes a normal subgroup of each group,
and <code>fScreen</code> 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 <code>Formation</code> can be used to obtain a
formation from the supplied library of formations. The formations
provided are:


<dl compact>
<dt><code>Formation( "Nilpotent" )</code> <dd> The formation of nilpotent groups,

<dt><code>Formation( "Supersolvable" )</code> <dd> The formation of supersolvable groups,

<dt><code>Formation( "Abelian" )</code> <dd> The formation of abelian groups,

<dt><code>Formation( "ElementaryAbelianProduct" )</code> <dd> The formation of direct products of elementary abelian groups,

<dt><code>Formation( "PNilpotent", prime )</code> <dd> The formation of p-nilpotent groups
for p = <code>prime</code>,

<dt><code>Formation( "PiGroups", primes )</code> <dd> The formation of π-groups for
π = the set <code>primes</code>,

<dt><code>Formation( "PLengthOne", prime )</code> <dd> The formation of groups of p-length 1
for p = <code>prime</code>.
</dl>

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<li><code>IsFormation( </code><var>F</var><code> ) C</code>
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<li><code>NameOfFormation( </code><var>F</var><code> ) A</code>
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<li><code>ResidualFunctionOfFormation( </code><var>F</var><code> ) A</code>

<code>IsFormation</code> returns <code>true</code> if and only if <var>F</var> is a GAP formation.
<code>NameOfFormation</code> returns the name of a formation and <code>ResidualFunctionOfFormation</code>
returns the residual function of a formation.

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<li><code>ScreenOfFormation( </code><var>F</var><code> ) A</code>

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

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<li><code>SupportOfFormation( </code><var>F</var><code> ) A</code>

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

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<li><code>ChangedSupport( </code><var>F</var><code>, </code><var>primes</var><code> ) O</code>

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

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<li><code>IsIntegrated( </code><var>F</var><code> ) P</code>

The local definition is called integrated in case F(p) is contained in
F for each prime p. The optional property <code>IsIntegrated</code> makes sense only if <code>HasScreenOfFormation( </code><var>F</var><code> )</code> is <code>true</code>. Notice that
some of the functions described below will require that all of the attributes <code>HasScreenOfFormation(
</code><var>F</var><code> )</code>, <code>HasIsIntegrated( </code><var>F</var><code> )</code> and <code>IsIntegrated( </code><var>F</var><code> )</code> are <code>true</code>. If
unbound, this property can be bound with <code>SetIsIntegrated</code>, but it is up to the
user to determine whether such a setting is appropriate.
Section <a href="CHAP006.htm">Formation Examples</a> contains an example of such usage.

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<li><code>Integrated( </code><var>F</var><code> ) O</code>

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 <var>F</var> be a locally defined GAP formation with
name <code></code><var>name</var><code></code>. If <var>F</var> is already integrated, then <code>Integrated( </code><var>F</var><code> )</code>
yields <var>F</var> itself. Otherwise, it yields a formation <code></code><var>name</var><code>Int</code> that is
abstractly the same as <var>F</var> but has integrated local definition.

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<li><code></code><var>F1</var><code> = </code><var>F2</var><code></code>
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<li><code></code><var>F1</var><code> < </code><var>F2</var><code></code>

Two GAP formations <var>F1</var> and <var>F2</var> 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.

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<li><code>Intersection( </code><var>F1</var><code>, </code><var>F2</var><code> ) O</code>

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

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<li><code>ProductOfFormations( </code><var>F1</var><code>, </code><var>F2</var><code> ) O</code>

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

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