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## compl.gd CRISP Burkhard Höfling
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## Copyright © 2000, 2002 Burkhard Höfling
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#F PcgsComplementOfChiefFactor(<pcgs>, <hpcgs>, <first>, <npcgs>, <kpcgs>)
##
## The arguments of PcgsComplementOfChiefFactor represent the following
## situation. Let H be a group, K < N < R, such that N/K is a
## H p-chief factor of H, and R is a normal subgroup of H which does
## : \ not centralise H/K, and such that R/N is elementary abelian
## ? R of exponent q (<> p).
## : / \
## Q N hpcgs is a pc sequence (i.e, a list of elements forming a pcgs,
## \ / but not necessarily a modulo pcgs) representing the factor
## K group H/N, such that hpcgs{[first..Length (hpcgs)]} represents
## R/N. npcgs is a modulo pcgs representing N/K. kpcgs is a pc
## sequence which generates K. All pc sequences and pcgses above must be
## induced with respect to pcgs, that is, the depths wrt. pcgs of their
## elements must be strictly increasing. Moreover, the depths (wrt pcgs) of
## the elementsin kpcgs must be strictly larger than the depths of the
## elements in hpcgs.
##
## PcgsComplementOfChiefFactor returns a pcgs (induced wrt pcgs) for a
## complement C of N/K in H. C is computed as the normaliser of Q, where
## Q/K is a Sylow q-subgroup of R/K, which will be computed in the course
## of the algorithm.
##
DeclareGlobalFunction("PcgsComplementOfChiefFactor");
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#F COMPLEMENT_SOLUTION_FUNCTION
##
## function used to compute a particular invariant complement
DeclareGlobalFunction("COMPLEMENT_SOLUTION_FUNCTION");
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#F ExtendedPcgsComplementsOfCentralModuloPcgsUnderAction(
## <act>, <pcgs>, <gpcgs>, <npcgs>, <kpcgs>, <all>)
##
## Let G be a group and <act> a set which acts on G via the caret operator.
## Moreover, let N and K be normal subgroups of G which are invariant under
## <act>, and assume that N/K is central in G. if <all> is true,
## ExtendedPcgsComplementsOfCentralModuloPcgsUnderAction computes the set
## of all normal subgroups C of G such that C/K complements N/K in G/K.
## If <all> is false, only one such complement is computed.
##
## <pcgs> is the pcgs wrt. to which all computations are carried out.
## <gpcgs> and <npcgs> are modulo pcgs of G/N and N/K, respectively,
## induced by <pcgs>, and <kpcgs> is a pcgs of K induced by <pcgs>.
##
## ExtendedPcgsComplementsOfCentralModuloPcgsUnderAction returns a
## record <rec> with components nrSolutions and solutionFunction.
## Each call <rec>.solutionFunction(<rec>, n) with an integer n
## with 1 <= n <= <rec>.nrSolutions gives the pcgs (induced wrt. to
## <pcgs>) of one possible subgroup C.
##
DeclareGlobalFunction("ExtendedPcgsComplementsOfCentralModuloPcgsUnderAction");
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#F PcgsInvariantComplementsOfElAbModuloPcgs(
## <act>, <numpcgs>, <pcgs>, <mpcgs>, <denpcgs>, <all>)
##
## computes invariant complements of the elementary abelian section
## N/L in G/L which are invariant under act.
## N/L is represented by the modulo pcgs <mpcgs>, G/N is in <pcgs>,
## G is in <numpcgs>, <L> is in <denpcgs>.
## If all is true, all such complements are computed, otherwise just one.
## If no complement exists, an empty list is returned.
##
DeclareGlobalFunction("PcgsInvariantComplementsOfElAbModuloPcgs");
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#F PcgsComplementsOfCentralModuloPcgsUnderActionNC(<act>,<pcgs>, <mpcgs>,<all>)
##
## similar to PcgsInvariantComplementsOfElAbModuloPcgs, except that it
## presumes that pcgs centralises mpcgs (and will probably produce an
## error if not).
##
DeclareGlobalFunction("PcgsComplementsOfCentralModuloPcgsUnderActionNC");
# [IsListOrCollection, IsModuloPcgs, IsModuloPcgs, IsBool]);
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#O InvariantComplementsOfElAbSection(<act>,<G>,<N>,<L>,<all>)
##
## computes complements of N/L in G/L which are invariant under act.
## act can be a collection of elements of a supergroup of G, or a collection
## of automorphisms of G which must fix G, L, N; however this is not checked.
## If all is true, all such complements are computed, otherwise just one.
## If no complement exists, or if N/L is not central in G/L, an empty
## list is returned if all is true, and fail is returned if all is false.
##
DeclareOperation("InvariantComplementsOfElAbSection",
[IsListOrCollection, IsGroup, IsGroup, IsGroup, IsBool]);
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#F ComplementsOfCentralSectionUnderAction(<act>,<G>,<N>,<L>,<all>)
#O ComplementsOfCentralSectionUnderActionNC(<act>,<G>,<N>,<L>,<all>)
##
## similar to ComplementsOfElAbSectionUnderAction; however G is expected
## to act centrally on N/L. ComplementsOfCentralSectionUnderActionNC
## does not check this.
##
DeclareGlobalFunction("ComplementsOfCentralSectionUnderAction");
DeclareOperation("ComplementsOfCentralSectionUnderActionNC",
[IsListOrCollection, IsGroup, IsGroup, IsGroup, IsBool]);
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#F ComplementsMaximalUnderAction(<act>, <ser>, <i>, <j>, <k>, <all>)
##
## computes subgroups C of ser[i] such that C/ser[k] is a act-invariant
## complement of ser[j]/ser[k] in ser[i]/ser[k], where i <= j <= k.
##
## ser[k]/ser[k] < ser[k-1]/ser[k] < ... < ser[i]/ser[k]
## must be a act-composition series of ser[i]/ser[k]. act must induce
## all inner automorphisms on ser[i].
##
## If all is true, it returns a list containing all such C.
## Otherwise it returns one C if it exists, or fail if no such C exists.
##
DeclareGlobalFunction("ComplementsMaximalUnderAction");
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#F PcgsComplementsMaximalUnderAction(<act>, <U>, <ser>, <j>, <k>, <all>)
##
## does the nontrivial work for ComplementsMaximalUnderAction above
##
DeclareGlobalFunction("PcgsComplementsMaximalUnderAction");
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#E
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