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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Alexander Hulpke.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains declarations for Morpheus
##
DeclareInfoClass("InfoMorph");
#############################################################################
##
#A AutomorphismGroup(<obj>)
##
## <#GAPDoc Label="AutomorphismGroup">
## <ManSection>
## <Attr Name="AutomorphismGroup" Arg='G'/>
##
## <Description>
## returns the full automorphism group of the group <A>G</A>.
## The automorphisms act on <A>G</A> by the caret operator <C>^</C>.
## The automorphism group often stores a <Ref Attr="NiceMonomorphism"/>
## value whose image is a permutation group,
## obtained by the action on a subset of <A>G</A>.
## <P/>
## Note that current methods for the calculation of the automorphism group
## of a group <M>G</M> require <M>G</M> to be a permutation group or
## a pc group to be efficient. For groups in other representations the
## calculation is likely very slow.
## <P/>
## Also, the <Package>AutPGrp</Package> package installs enhanced methods
## for <Ref Attr="AutomorphismGroup"/> for finite <M>p</M>-groups, and
## the <Package>FGA</Package> package - for finitely generated subgroups
## of free groups.
## <P/>
## Methods may be installed for <Ref Attr="AutomorphismGroup"/>
## for other domains, such as e.g. for linear codes in the
## <Package>GUAVA</Package> package, loops in the <Package>loops</Package>
## package and nilpotent Lie algebras in the <Package>Sophus</Package>
## package (see package manuals for their descriptions).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("AutomorphismGroup",IsDomain);
#############################################################################
##
#P IsGroupOfAutomorphisms(<G>)
##
## <#GAPDoc Label="IsGroupOfAutomorphisms">
## <ManSection>
## <Prop Name="IsGroupOfAutomorphisms" Arg='G'/>
##
## <Description>
## indicates whether <A>G</A> consists of automorphisms of another group
## <M>H</M>.
## The group <M>H</M> can be obtained from <A>G</A> via the attribute
## <Ref Attr="AutomorphismDomain"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsGroupOfAutomorphisms", IsGroup );
InstallTrueMethod( IsGroup, IsGroupOfAutomorphisms );
#############################################################################
##
#P IsGroupOfAutomorphismsFiniteGroup(<G>)
##
## <ManSection>
## <Prop Name="IsGroupOfAutomorphismsFiniteGroup" Arg='G'/>
##
## <Description>
## indicates whether <A>G</A> consists of automorphisms of another finite group <A>H</A>.
## The group <A>H</A> can be obtained from <A>G</A> via the attribute
## <C>AutomorphismDomain</C>.
## </Description>
## </ManSection>
##
DeclareProperty( "IsGroupOfAutomorphismsFiniteGroup", IsGroup );
InstallTrueMethod( IsGroup, IsGroupOfAutomorphismsFiniteGroup );
InstallTrueMethod( IsGroupOfAutomorphisms,IsGroupOfAutomorphismsFiniteGroup);
InstallTrueMethod( IsFinite,IsGroupOfAutomorphismsFiniteGroup);
InstallTrueMethod( IsHandledByNiceMonomorphism,
IsGroupOfAutomorphismsFiniteGroup);
InstallSubsetMaintenance( IsGroupOfAutomorphisms,
IsGroup and IsGroupOfAutomorphisms, IsGroup );
InstallSubsetMaintenance( IsGroupOfAutomorphismsFiniteGroup,
IsGroup and IsGroupOfAutomorphismsFiniteGroup, IsGroup );
#############################################################################
##
#A AutomorphismDomain(<G>)
##
## <#GAPDoc Label="AutomorphismDomain">
## <ManSection>
## <Attr Name="AutomorphismDomain" Arg='G'/>
##
## <Description>
## If <A>G</A> consists of automorphisms of <M>H</M>,
## this attribute returns <M>H</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "AutomorphismDomain", IsGroupOfAutomorphisms );
#############################################################################
##
#P IsAutomorphismGroup(<G>)
##
## <#GAPDoc Label="IsAutomorphismGroup">
## <ManSection>
## <Prop Name="IsAutomorphismGroup" Arg='G'/>
##
## <Description>
## indicates whether <A>G</A>, which must be
## <Ref Prop="IsGroupOfAutomorphisms"/>,
## is the full automorphism group of another
## group <M>H</M>, this group is given as <Ref Attr="AutomorphismDomain"/>
## value of <A>G</A>.
## <Example><![CDATA[
## gap> g:=Group((1,2,3,4),(1,3));
## Group([ (1,2,3,4), (1,3) ])
## gap> au:=AutomorphismGroup(g);
## <group of size 8 with 3 generators>
## gap> GeneratorsOfGroup(au);
## [ Pcgs([ (2,4), (1,2,3,4), (1,3)(2,4) ]) ->
## [ (1,2)(3,4), (1,2,3,4), (1,3)(2,4) ],
## Pcgs([ (2,4), (1,2,3,4), (1,3)(2,4) ]) ->
## [ (1,3), (1,2,3,4), (1,3)(2,4) ],
## Pcgs([ (2,4), (1,2,3,4), (1,3)(2,4) ]) ->
## [ (2,4), (1,4,3,2), (1,3)(2,4) ] ]
## gap> NiceObject(au);
## Group([ (1,2,4,6), (1,4)(2,6), (2,6)(3,5) ])
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsAutomorphismGroup", IsGroupOfAutomorphisms );
InstallTrueMethod( IsGroupOfAutomorphisms,IsAutomorphismGroup );
#############################################################################
##
#A InnerAutomorphismsAutomorphismGroup(<autgroup>)
##
## <#GAPDoc Label="InnerAutomorphismsAutomorphismGroup">
## <ManSection>
## <Attr Name="InnerAutomorphismsAutomorphismGroup" Arg='autgroup'/>
##
## <Description>
## For an automorphism group <A>autgroup</A> of a group
## this attribute stores the subgroup of inner automorphisms
## (automorphisms induced by conjugation) of the original group.
## <Example><![CDATA[
## gap> InnerAutomorphismsAutomorphismGroup(au);
## <group with 2 generators>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("InnerAutomorphismsAutomorphismGroup",IsGroup);
#############################################################################
##
#A InnerAutomorphismGroup(<G>)
##
## <#GAPDoc Label="InnerAutomorphismGroup">
## <ManSection>
## <Attr Name="InnerAutomorphismGroup" Arg='G'/>
##
## <Description>
## For a group <A>G</A> this attribute stores the group of inner
## automorphisms (automorphisms induced by conjugation) of the original group.
## <Example><![CDATA[
## gap> InnerAutomorphismGroup(SymmetricGroup(5));
## <group with 2 generators>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("InnerAutomorphismGroup", IsGroup);
#############################################################################
##
#F AssignNiceMonomorphismAutomorphismGroup(<autgrp>,<group>) local
##
## <#GAPDoc Label="AssignNiceMonomorphismAutomorphismGroup">
## <ManSection>
## <Func Name="AssignNiceMonomorphismAutomorphismGroup" Arg='autgrp, group'/>
##
## <Description>
## computes a nice monomorphism for <A>autgroup</A> acting on <A>group</A>
## and stores it as <Ref Attr="NiceMonomorphism"/> value of <A>autgrp</A>.
## <P/>
## If the centre of <Ref Attr="AutomorphismDomain"/> of <A>autgrp</A> is
## trivial, the operation will first try to represent all automorphisms by
## conjugation (in <A>group</A> or in a natural parent of <A>group</A>).
## <P/>
## If this fails the operation tries to find a small subset of <A>group</A>
## on which the action will be faithful.
## <P/>
## The operation sets the attribute <Ref Attr="NiceMonomorphism"/>
## and does not return a value.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("AssignNiceMonomorphismAutomorphismGroup");
#############################################################################
##
#F NiceMonomorphismAutomGroup(<autgrp>,<elms>,<elmsgens>)
##
## <#GAPDoc Label="NiceMonomorphismAutomGroup">
## <ManSection>
## <Func Name="NiceMonomorphismAutomGroup" Arg='autgrp, elms, elmsgens'/>
##
## <Description>
## This function creates a monomorphism for an automorphism group
## <A>autgrp</A> of a group by permuting the group elements in the list
## <A>elms</A>.
## This list must be chosen to yield a faithful representation.
## <A>elmsgens</A> is a list of generators which are a subset of
## <A>elms</A>.
## (They can differ from the group's original generators.)
## It does not yet assign it as <Ref Attr="NiceMonomorphism"/> value.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("NiceMonomorphismAutomGroup");
#############################################################################
##
#F MorFroWords(<gens>) . . . . . . create some pseudo-random words in <gens>
##
## <ManSection>
## <Func Name="MorFroWords" Arg='gens'/>
##
## <Description>
## This function takes a generator list <A>gens</A> and creates a list of
## pseudo-random words in them. These words can be used for example to test
## quickly whether generator mappings extend to a homomorphism. The words
## are taken from the MeatAxe FRO routine.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("MorFroWords");
#############################################################################
##
#F MorRatClasses(<G>) . . . . . . . . . . . local
##
## <ManSection>
## <Func Name="MorRatClasses" Arg='G'/>
##
## <Description>
## yields a list of rational classes as a collection of ordinary classes.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("MorRatClasses");
#############################################################################
##
#F MorMaxFusClasses(<l>) . . maximal possible morphism fusion of classlists
##
## <ManSection>
## <Func Name="MorMaxFusClasses" Arg='l'/>
##
## <Description>
## computes a list of classes (as unions of rational classes) which will be
## respected by any automorphism. This is used to determine potential
## automorphism images of elements.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("MorMaxFusClasses");
#############################################################################
##
#F MorClassLoop(<range>,<classes>,<params>,<action>) class loop
##
## <#GAPDoc Label="MorClassLoop">
## <ManSection>
## <Func Name="MorClassLoop" Arg='range, classes, params, action'/>
##
## <Description>
## This function loops over element tuples taken from <A>classes</A> and
## checks these for properties such as generating a given group,
## or fulfilling relations.
## This can be used to find small generating sets or all types of Morphisms.
## The element tuples are used only up to inner automorphisms as
## all images can be obtained easily from them by conjugation while
## running through all of them usually would take too long.
## <P/>
## <A>range</A> is a group from which these elements are taken.
## The classes are given in a list <A>classes</A> which is a list of records
## with the following components.
## <List>
## <Mark><C>classes</C></Mark>
## <Item>
## list of conjugacy classes
## </Item>
## <Mark><C>representative</C></Mark>
## <Item>
## One element in the union of these classes
## </Item>
## <Mark><C>size</C></Mark>
## <Item>
## The sum of the sizes of these classes
## </Item>
## </List>
## <P/>
## <A>params</A> is a record containing the following optional components.
## <List>
## <Mark><C>gens</C></Mark>
## <Item>
## generators that are to be mapped (for testing morphisms). The length
## of this list determines the length of element tuples considered.
## </Item>
## <Mark><C>from</C></Mark>
## <Item>
## a preimage group (that contains <C>gens</C>)
## </Item>
## <Mark><C>to</C></Mark>
## <Item>
## image group (which might be smaller than <C>range</C>)
## </Item>
## <Mark><C>free</C></Mark>
## <Item>
## free generators, a list of the same length than the
## generators <C>gens</C>.
## </Item>
## <Mark><C>rels</C></Mark>
## <Item>
## some relations that hold among the generators <C>gens</C>.
## They are given as a list <C>[ word, order ]</C>
## where <C>word</C> is a word in the free generators <C>free</C>.
## </Item>
## <Mark><C>dom</C></Mark>
## <Item>
## a set of elements on which automorphisms act faithfully (used to do
## element tests in partial automorphism groups).
## </Item>
## <Mark><C>aut</C></Mark>
## <Item>
## Subgroup of already known automorphisms.
## </Item>
## <Mark><C>condition</C></Mark>
## <Item>
## A function that will be applied to the homomorphism and must return
## <K>true</K> for the homomorphism to be accepted.
## </Item>
## </List>
## <P/>
## <A>action</A> is a number whose bit-representation indicates
## the requirements which are enforced on the element tuples found,
## as follows.
## <List>
## <Mark>1</Mark>
## <Item>
## homomorphism
## </Item>
## <Mark>2</Mark>
## <Item>
## injective
## </Item>
## <Mark>4</Mark>
## <Item>
## surjective
## </Item>
## <Mark>8</Mark>
## <Item>
## find all (otherwise stops after the first find)
## </Item>
## </List>
## If the search is for homomorphisms, the function returns homomorphisms
## obtained by mapping the given generators <C>gens</C>
## instead of element tuples.
## <P/>
## The <Q>Morpheus</Q> algorithm used to find homomorphisms is described in
## <Cite Key="Hulpke96" Where="Section V.5"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("MorClassLoop");
#############################################################################
##
#F MorFindGeneratingSystem(<G>,<cl>) . . local
##
## <ManSection>
## <Func Name="MorFindGeneratingSystem" Arg='G,cl'/>
##
## <Description>
## tries to find generating system with as few as possible generators
## which will be taken preferraby from the first classes in <A>cl</A>
## </Description>
## </ManSection>
##
DeclareGlobalFunction("MorFindGeneratingSystem");
#############################################################################
##
#F Morphium(<G>,<H>,<DoAuto>) . . . . . . . . local
##
## <ManSection>
## <Func Name="Morphium" Arg='G,H,DoAuto'/>
##
## <Description>
## This function is a frontend to <C>MorClassLoop</C> and is used to find
## isomorphisms between <A>G</A> and <A>H</A> or the automorphism group of <A>G</A> (in which
## case <A>G</A> must equal <A>H</A>). The boolean flag <A>DoAuto</A> indicates if all
## automorphisms should be found.
## The function requires, that both groups are not cyclic!
## </Description>
## </ManSection>
##
DeclareGlobalFunction("Morphium");
#############################################################################
##
#F AutomorphismGroupAbelianGroup(<G>)
##
## <ManSection>
## <Func Name="AutomorphismGroupAbelianGroup" Arg='G'/>
##
## <Description>
## computes the automorphism group of an abelian group <A>G</A>, using the theorem
## of Shoda.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("AutomorphismGroupAbelianGroup");
DeclareGlobalFunction("AutomorphismGroupFittingFree");
#############################################################################
##
#F IsomorphismAbelianGroups(<G>,<H>)
##
## <ManSection>
## <Func Name="IsomorphismAbelianGroups" Arg='G,H'/>
##
## <Description>
## computes an isomorphism between the abelian groups <A>G</A> and <A>H</A>
## if they are isomorphic and returns <K>fail</K> otherwise.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("IsomorphismAbelianGroups");
#############################################################################
##
#F IsomorphismGroups(<G>,<H>)
##
## <#GAPDoc Label="IsomorphismGroups">
## <ManSection>
## <Func Name="IsomorphismGroups" Arg='G,H'/>
##
## <Description>
## computes an isomorphism between the groups <A>G</A> and <A>H</A>
## if they are isomorphic and returns <K>fail</K> otherwise.
## <P/>
## With the existing methods the amount of time needed grows with
## the size of a generating system of <A>G</A>. (Thus in particular for
## <M>p</M>-groups calculations can be slow.) If you do only need to know
## whether groups are isomorphic, you might want to consider
## <Ref BookName="smallgrp" Func="IdGroup"/> or the random isomorphism test
## (see <Ref Func="RandomIsomorphismTest"/>).
## <P/>
## <Index Subkey="find all">isomorphisms</Index>
## <Example><![CDATA[
## gap> g:=Group((1,2,3,4),(1,3));;
## gap> h:=Group((1,4,6,7)(2,3,5,8), (1,5)(2,6)(3,4)(7,8));;
## gap> IsomorphismGroups(g,h);
## [ (1,2,3,4), (1,3) ] -> [ (1,4,6,7)(2,3,5,8), (1,2)(3,7)(4,8)(5,6) ]
## gap> IsomorphismGroups(g,Group((1,2,3,4),(1,2)));
## fail
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("IsomorphismGroups");
DeclareGlobalFunction("IsomorphismSimpleGroups");
#############################################################################
##
#O GQuotients(<F>,<G>) . . . . . epimorphisms from F onto G up to conjugacy
##
## <#GAPDoc Label="GQuotients">
## <ManSection>
## <Oper Name="GQuotients" Arg='F, G'/>
##
## <Description>
## computes all epimorphisms from <A>F</A> onto <A>G</A> up to automorphisms
## of <A>G</A>.
## This classifies all factor groups of <A>F</A> which are isomorphic to
## <A>G</A>.
## <P/>
## With the existing methods the amount of time needed grows with
## the size of a generating system of <A>G</A>. (Thus in particular for
## <M>p</M>-groups calculations can be slow.)
## <P/>
## If the <C>findall</C> option is set to <K>false</K>,
## the algorithm will stop once one homomorphism has been found
## (this can be faster and might be sufficient if not all homomorphisms
## are needed).
## <P/>
## <Index Subkey="find all">epimorphisms</Index>
## <Index Subkey="find all">projections</Index>
## <Example><![CDATA[
## gap> g:=Group((1,2,3,4),(1,2));
## Group([ (1,2,3,4), (1,2) ])
## gap> h:=Group((1,2,3),(1,2));
## Group([ (1,2,3), (1,2) ])
## gap> quo:=GQuotients(g,h);
## [ [ (2,4), (1,4,3) ] -> [ (2,3), (1,2,3) ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("GQuotients",[IsGroup,IsGroup]);
#############################################################################
##
#O IsomorphicSubgroups(<G>,<H>) monomorphisms from H onto G up to conjugacy
##
## <#GAPDoc Label="IsomorphicSubgroups">
## <ManSection>
## <Oper Name="IsomorphicSubgroups" Arg='G,H'/>
##
## <Description>
## computes all monomorphisms from <A>H</A> into <A>G</A> up to
## <A>G</A>-conjugacy of the image groups.
## This classifies all <A>G</A>-classes of subgroups of <A>G</A> which
## are isomorphic to <A>H</A>.
## <P/>
## With the existing methods, the amount of time needed grows with
## the size of a generating system of <A>G</A>. (Thus in particular for
## <M>p</M>-groups calculations can be slow.) A main use of
## <Ref Oper="IsomorphicSubgroups"/> therefore is to find nonsolvable
## subgroups (which often can be generated by 2 elements).
## <P/>
## (To find <M>p</M>-subgroups it is often faster to compute the subgroup
## lattice of the Sylow subgroup and to use <Ref BookName="smallgrp" Func="IdGroup"/>
## to identify the type of the subgroups.)
## <P/>
## If the <C>findall</C> option is set to <K>false</K>,
## the algorithm will stop once one homomorphism has been found
## (this can be faster and might be sufficient if not all homomorphisms are
## needed).
## <P/>
## <Index Subkey="find all">embeddings</Index>
## <Index Subkey="find all">monomorphisms</Index>
## <Example><![CDATA[
## gap> g:=Group((1,2,3,4),(1,2));
## Group([ (1,2,3,4), (1,2) ])
## gap> h:=Group((3,4),(1,2));;
## gap> emb:=IsomorphicSubgroups(g,h);
## [ [ (3,4), (1,2) ] -> [ (1,2), (3,4) ],
## [ (3,4), (1,2) ] -> [ (1,3)(2,4), (1,2)(3,4) ] ]
## gap> g1:=PSO(-1,8,2);;
## gap> Length(IsomorphicSubgroups(g1,PSL(2,7)));
## 3
## gap> Length(IsomorphicSubgroups(g1,PSL(2,7):findall:=false));
## 1
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("IsomorphicSubgroups",[IsGroup,IsGroup]);
DeclareGlobalFunction("PatheticIsomorphism");
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