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SSL forms.tex Sprache: Latech |
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%% %W forms.tex FORMAT documentation B. Eick and C.R.B. Wright %% %% 10-31-11 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Formations in GAP} %\index{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 propert smallest normal subgroup $G^{\F}$ with factor group $G / G^{\F}$ in~$\F$. The subgroup $G^{\F}$ 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 $G^{\F}$ for each (finite solvable) group $G$, and from that perspective $\F$ consists of the groups $G$ for which $G^{\F}$ 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 eac 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 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 $G^{\F(p)}$ 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 $G^{\F(p)}$, to test ``centrality.'' Section~"Other Applications" describes one s 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: \beginitems `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 $\pi$-groups for $\pi =$ the set `primes', `Formation( "PLengthOne", prime )' & The formation of groups of $p$-length 1 for $p =$ `prime'. \enditems \> 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 `ResidualFunctionOfFormati 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 `ScreenOfFormation( <F> )( <G>, <p> )' returns $G^{F(p)}$ if 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 `HasS some of the functions described below will require that all of the attributes `HasScreenOfFor <F> )', `HasIsIntegrated( 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 yields <F> itself. Otherwise, it yields a formation `<name>Int' that is abstractly the same as <F> but has integrated local definition. \> `<F1> = <F2>' {formation!equality} \> `<F1> \< <F2>' {formation!comparison} 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 $G^{F2}$ 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 pro `ProductOfFormations( <F1>, <F2> )' yields the product `( formations. The product has the attribute `ResidualFunctionOfFormation' and has the attribute `ScreenOfFormation' whenever both or whenever both `HasScreenOfFormation( <F2> )' and `not HasSupportOfFormation( <F1> )' are `true'. In these cases the property `IsIntegrated' will be inherited if possible. ¤ Dauer der Verarbeitung: 0.26 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28