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##
## projector.gd CRISP Burkhard Höfling
##
## Copyright © 2000 Burkhard Höfling
##
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##
#V InfoProjector
##
DeclareInfoClass("InfoProjector");
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##
#O Projector(<grp>, <class>)
##
KeyDependentOperation("Projector", IsGroup, IsGroupClass, ReturnTrue);
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##
#M CoveringSubgroup(<grp>, <class>)
##
KeyDependentOperation("CoveringSubgroup", IsGroup, IsGroupClass,
ReturnTrue);
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##
#O ProjectorFromExtendedBoundaryFunction(<grp>, <data>, <inonly>)
##
## if inonly is false, this computes a projector of <grp> for the
## Schunck class described by <data>.
## if inonly is true, it returns true or false depending whether <grp>
## belongs to the Schunck class or not.
##
## See PROJECTOR_FROM_BOUNDARY below for the meaning of <data>.
##
DeclareOperation("ProjectorFromExtendedBoundaryFunction",
[IsGroup, IsRecord, IsBool]);
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##
#F PROJECTOR_FROM_BOUNDARY(<gpcgs>, <data>, <inonly>, <hom>, <conv>)
##
## <gpcgs> is a pcgs of the group G for which the computation will be
## performed.
##
## If the boolean inonly is true, the function returns true if G belongs to
## H, and false otherwise. If inonly is false, it returns the pcgs(an
## induced pcgs wrt. gpcgs) of a projector of G.
##
## hom and conv are booleans. If hom is true, computations will be carried
## out in factor groups, otherwise wrt to modulo pcgs. If hom or conv is
## true, computations will take place wrt a pcgs refining an elementary
## abelian series of G, otherwise wrt. the pcgs <gpcgs> supplied.
##
## <data> must contain four components, dfunc, cfunc, kfunc, and bfunc,
## each bound to a function. The purpose of these functions is as follows.
##
## - data.dfunc takes four arguments, upcgs, npcgs, p, and data.
## Here upcgs is a pcgs of the group U and npcgs is a modulo pcgs
## induced by upcgs which represents a p-chief factor N/L of U. Moreover,
## U/N belongs to the Schunck class H. data is just the argument of
## PROJECTOR_FROM_BOUNDARY. data.dfunc may return true, false, or fail.
## data.dfunc may return true if U/L does not belong to H, and false if
## U/L belongs to H.(An example may be to use information on whether
## groups in H can have order divisible by p).
##
## - data.cfunc takes five arguments, upcgs, npcgs, p, cent, and data.
## The meaning of the arguments and the purpose of the function is the
## same as for data.dfunc, except there is one additional information
## available: the subgroup D of U represented by upcgs{[cent..
## Length(upcgs)]} centralises npcgs, and if D < U, then there exists a
## normal subgroup R of U such that R/D is elementary abelian and R does
## not centralise npcgs.(The obvious interpretation is that npcgs is
## central iff cent = 1). data.cfunc is only called when data.dfunc has
## returned fail.
##
## - data.kfunc has six arguments: upcgs, kpcgs, npcgs, p, cent, and data.
## Except for kpcgs, the meaning is the same as for cfunc and dfunc.
## In addition, kpcgs is a pcgs induced from upcgs for a normal subgroup K
## of U such that K/L is a complement of N/L in D/L. Note that the
## existence of K implies that K/L is a complemented chief factor of U/L
## (see the accompanying article "crisp.dvi" for details). data.kfunc is
## only called when cfunc and dfunc have both returned fail, and if K
## exists. One possible application is that if H is a local formation
## defined by a formation function f, then U/L belongs to H if and only if
## the f(p)-residual of U centralises N/L.
##
## - data.bfunc has arguments upcgs, cpcgs, kpcgs, npcgs, p, cent, and
## data. Except for cpcgs, they are the same as above. bfunc
## U must either return true or false. it is only called if all
## / \ of the above functions have returned fail. cpcgs is a pcgs
## C R induced from upcgs for a maximal subgroup C of U which
## \ \ complements N/L and contains K. In this situation, it is
## \ D known that U/Core_U(C) is a primitive group with socle
## \ / \ N Core_U(C)/Core_U(C) which is U-isomorphic with N/L;
## K N moreover Core_U(C) = C_C(N/L) contains K. Also, U/L belongs
## \ / to H if and only if U/Core_U(C) belongs to(the basis of) H,
## L and otherwise U/Core_U(C) is in the boundary of H. Therefore
## information about the basis or boundary of H is sufficient
## for the test to be performed by bfunc.
##
## Note that it is a good idea only to perform cheap tests by data.dfunc and
## data.cfunc, and leave expensive tests to data.kfunc and data.bfunc,
## because in that case, the expensive tests are only carried out if it is
## known that N/L is complemented in U.(Otherwise N/L is a Frattini chief
## factor, and U/L must belong to H - no further test is required.)
##
DeclareGlobalFunction("PROJECTOR_FROM_BOUNDARY");
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##
#F DFUNC_FROM_CHARACTERISTIC(<upcgs>, <npcgs>, <p>, <data>)
##
## standard function to pass to PcgsProjectorFromExtendedBoundaryFunction
## as data.dfunc, where the argument data must have a component char
## containing the characteristic of the formation. Every prime divisor of
## any group in the class must be in the characteristic.
##
DeclareGlobalFunction("DFUNC_FROM_CHARACTERISTIC");
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##
#F DFUNC_FROM_MEMBER_FUNCTION(<upcgs>, <npcgs>, <p>, <data>)
##
## standard function to pass to PcgsProjectorFromExtendedBoundaryFunction
## as data.cfunc, where the argument data must have a component memberf
## containing a function which decides membership in the Schunck class.
##
DeclareGlobalFunction("DFUNC_FROM_MEMBER_FUNCTION");
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##
#F CFUNC_FROM_CHARACTERISTIC(<upcgs>, <npcgs>, <p>, <centind>, <data>)
##
## standard function to pass to PcgsProjectorFromExtendedBoundaryFunction
## as data.cfunc, where the argument data must have a component char
## containing the characteristic of the class. Every prime divisor of any
## group in the class must be in the characteristic. Otherwise
## CFUNC_FROM_CHARACTERISTIC_SCHUNCK may be used.
##
DeclareGlobalFunction("CFUNC_FROM_CHARACTERISTIC");
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##
#F KFUNC_FROM_LOCAL_DEFINITION(<upcgs>, <kpcgs>, <npcgs>, <p>,
## <centind>, <data>)
##
## standard function to pass to PcgsProjectorFromExtendedBoundaryFunction
## as data.kfunc. The argument data must have a component lfunc containing
## a function taking three arguments: a group G, a prime p, and a record r.
## It must return a list of elements of G such that the smallest normal
## subgroup of G containing them is the f(p)-residual of G, or fail if
## f(p) is empty, where f is a local function for the formation for which
## the computation should be carried out. The argument data passed to
## PcgsProjectorFromExtendedBoundaryFunction will be in r when lfunc is
## called.
##
DeclareGlobalFunction("KFUNC_FROM_LOCAL_DEFINITION");
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##
#F CFUNC_FROM_CHARACTERISTIC_SCHUNCK(<upcgs>, <npcgs>, <p>,
## <centind>, <data>)
##
## standard function to pass to PcgsProjectorFromExtendedBoundaryFunction
## as data.dfunc, where the argument data must have a component char
## containing the characteristic of the Schunck class
##
DeclareGlobalFunction("CFUNC_FROM_CHARACTERISTIC_SCHUNCK");
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##
#F BFUNC_FROM_TEST_FUNC_FAC(<upcgs>, <cpcgs>, <kpcgs>, <npcgs>, <p>,
## <centind>, <data>)
##
## standard function to pass to PcgsProjectorFromExtendedBoundaryFunction
## as data.bfunc. The argument data must have a component test containing
## a function taking two arguments, a primitive soluble group G such that
## G/Socle(G) belongs to the Schunck class H in question, and a record r.
## data.test must return true if G is in the boundary of the Schunck class
## H, and false if it belongs to H.
##
DeclareGlobalFunction("BFUNC_FROM_TEST_FUNC_FAC");
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##
#F BFUNC_FROM_TEST_FUNC_MOD(<upcgs>, <cpcgs>, <kpcgs>, <npcgs>, <p>,
## <centind>, <data>)
##
## this is the same as BFUNC_FROM_TEST_FUNC, except that it uses
## CentralizerModulo to compute the centralizer of a chief factor.
##
## In ProjectorOp, we may want to use this function, rather than
## BFUNC_FROM_TEST_FUNC_FAC because its performance seems to be the
## same for pc groups, while it might work better for perm groups because
## one does not have to work in factor groups.
##
DeclareGlobalFunction("BFUNC_FROM_TEST_FUNC_MOD");
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##
#F BFUNC_FROM_TEST_FUNC(<upcgs>, <cpcgs>, <kpcgs>, <npcgs>, <p>,
## <centind>, <data>)
##
## Presently, we only use BFUNC_FROM_TEST_FUNC_FAC because of a bug in
## CentralizerModulo in the released version (4.2 fix 5) of GAP.
##
DeclareSynonym("BFUNC_FROM_TEST_FUNC", BFUNC_FROM_TEST_FUNC_FAC);
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##
#E
##
