<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<!-- %% -->
<!-- %A  grpperm.xml                 GAP documentation            Alexander Hulpke -->
<!-- %A                                                           Heiko Theißen -->
<!-- %A                                                           Ákos Seress -->
<!-- %% -->
<!-- %% -->
<!-- %Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland -->
<!-- %Y  Copyright (C) 2002 The GAP Group -->
<!-- %% -->
<Chapter Label="Permutation Groups">
<Heading>Permutation Groups</Heading>

<!-- %%  The code for the computation and verification stabilizer chains, for -->
<!-- %%  composition series and the radical is due to \Ákos Seress. -->
<!-- %%  Heiko Theißen wrote the backtrack routines which are used to compute for -->
<!-- %%  example centralizers, normalizers and conjugating elements. -->


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:IsPermGroup">
<Heading>IsPermGroup (Filter)</Heading>

<#Include Label="IsPermGroup">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="The Natural Action">
<Heading>The Natural Action</Heading>

The functions
<Ref Attr="MovedPoints" Label="for a list or collection of permutations"/>,
<Ref Attr="NrMovedPoints" Label="for a list or collection of permutations"/>,
<Ref Attr="LargestMovedPoint" Label="for a list or collection of permutations"/>,
and
<Ref Attr="SmallestMovedPoint" Label="for a list or collection of permutations"/>
are defined for arbitrary collections of
permutations (see&nbsp;<Ref Sect="Moved Points of Permutations"/>),
in particular they can be applied to permutation groups.
<Example><![CDATA[
gap> g:= Group( (2,3,5,6), (2,3) );;
gap> MovedPoints( g );  NrMovedPoints( g );
[ 2, 3, 5, 6 ]
4
gap> LargestMovedPoint( g );  SmallestMovedPoint( g );
6
2
]]></Example>
<P/>
The action of a permutation group on the positive integers is a group
action (via the acting function <Ref Func="OnPoints"/>).
Therefore all action functions can be applied
(see the Chapter&nbsp;<Ref Chap="Group Actions"/>),
for example <Ref Oper="Orbit"/>, <Ref Func="Stabilizer"/>,
<Ref Oper="Blocks" Label="for a group, an action domain, etc."/>,
<Ref Oper="IsTransitive" Label="for a group, an action domain, etc."/>,
<Ref Oper="IsPrimitive" Label="for a group, an action domain, etc."/>.
<P/>
If one has a list of group generators and is interested in the moved points
(see above) or orbits, it may be useful to avoid the explicit construction
of the group for efficiency reasons.
For the special case of the action of permutations on positive integers
via <C>^</C>,
the functions <Ref Func="OrbitPerms"/> and <Ref Func="OrbitsPerms"/>
are provided for this purpose.
<P/>
Similarly, several functions concerning the natural action of
permutation groups address stabilizer chains
(see&nbsp;<Ref Sect="Stabilizer Chains"/>)
rather than permutation groups themselves,
for example <Ref Func="BaseStabChain"/>.

<#Include Label="OrbitPerms">
<#Include Label="OrbitsPerms">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Computing a Permutation Representation">
<Heading>Computing a Permutation Representation</Heading>

<#Include Label="IsomorphismPermGroup">
<#Include Label="SmallerDegreePermutationRepresentation">

<Example><![CDATA[
gap> p:=Group((1,2,3,4,5,6),(1,2));;p:=Action(p,AsList(p),OnRight);;
gap> Length(MovedPoints(p));
720
gap> q:=SmallerDegreePermutationRepresentation(p);;
gap> NrMovedPoints(Image(q));
6
]]></Example>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Symmetric and Alternating Groups">
<Heading>Symmetric and Alternating Groups</Heading>

The commands <Ref Func="SymmetricGroup" Label="for a degree"/> and
<Ref Func="AlternatingGroup" Label="for a degree"/>
(see Section&nbsp;<Ref Sect="Basic Groups"/>)
construct symmetric and alternating permutation groups.
&GAP; can also detect whether a given permutation group is a symmetric
or alternating group on the set of its moved points;
if so then the group is called a <E>natural</E> symmetric or alternating
group, respectively.
<P/>
The functions <Ref Prop="IsSymmetricGroup"/> and
<Ref Prop="IsAlternatingGroup"/> can be used to check whether a given group
(not necessarily a permutation group)
is isomorphic to a symmetric or alternating group.
<P/>

<#Include Label="IsNaturalSymmetricGroup">
<#Include Label="IsSymmetricGroup">
<#Include Label="IsAlternatingGroup">
<#Include Label="SymmetricParentGroup">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Primitive Groups">
<Heading>Primitive Groups</Heading>

<#Include Label="ONanScottType">
<#Include Label="SocleTypePrimitiveGroup">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Stabilizer Chains">
<Heading>Stabilizer Chains</Heading>

Many of the algorithms for permutation groups use a <E>stabilizer chain</E>
of the group.
The concepts of stabilizer chains, <E>bases</E>,
and <E>strong generating sets</E> were
introduced by Charles Sims in&nbsp;<Cite Key="Sim70"/>. An extensive account
of basic algorithms together with asymptotic runtime analysis can be found in
reference&nbsp;<Cite Key="Seress2003" Where="Chapter 4"/>.
A further discussion of base change is given in
section&nbsp;<Ref Sect="Generalized Conjugation Technique"/>.
<P/>
Let <M>B = [ b_1, \ldots, b_n ]</M> be a list of points,
<M>G^{(1)} = G</M> and <M>G^{{(i+1)}} = Stab_{{G^{(i)}}}(b_i)</M>,
such that <M>G^{(n+1)} = \{ () \}</M>.
Then the list <M>[ b_1, \ldots, b_n ]</M> is called a <E>base</E> of <M>G</M>,
the points <M>b_i</M> are called <E>base points</E>.
A set <M>S</M> of generators for <M>G</M> satisfying the condition
<M>\langle S \cap G^{(i)} \rangle = G^{(i)}</M> for each
<M>1 \leq i \leq n</M>,
is called a <E>strong generating set</E> (SGS) of <M>G</M>.
(More precisely we ought to say that it is a SGS of <M>G</M> <E>relative</E>
to <M>B</M>).
The chain of subgroups <M>G^{(i)}</M> of <M>G</M> itself is
called the <E>stabilizer chain</E> of <M>G</M> relative to <M>B</M>.
<P/>
Since <M>[ b_1, \ldots, b_n ]</M>, where <M>n</M> is the degree of <M>G</M>
and <M>b_i</M> are the moved points of <M>G</M>,
certainly is a base for <M>G</M> there exists a base for
each permutation group.
The number of points in a base is called the <E>length</E> of the base.
A base <M>B</M> is called <E>reduced</E> if there exists no <M>i</M>
such that <M>G^{(i)} = G^{(i+1)}</M>.
(This however does not imply that no subset of <M>B</M> could also serve
as a base.)
Note that different reduced bases for one permutation group <M>G</M>
may have different lengths.
For example, the irreducible degree <M>416</M> permutation representation
of the Chevalley Group <M>G_2(4)</M> possesses reduced bases of lengths
<M>5</M> and <M>7</M>.
<P/>
Let <M>R^{(i)}</M> be a right transversal of <M>G^{(i+1)}</M> in
<M>G^{(i)}</M>, i.e. a set of right coset representatives of the cosets of
<M>G^{(i+1)}</M> in <M>G^{(i)}</M>.
Then each element <M>g</M> of <M>G</M> has a unique representation as a
product of the form <M>g = r_n \ldots r_1</M> with <M>r_i \in R^{(i)}</M>.
The cosets of <M>G^{(i+1)}</M> in <M>G^{(i)}</M> are in bijective
correspondence with the points in <M>O^{(i)} := b_i^{{G^{(i)}}}</M>.
So we could represent a transversal as a list <M>T</M> such that <M>T[p]</M>
is a representative of the coset corresponding to the point
<M>p \in O^{(i)}</M>,
i.e., an element of <M>G^{(i)}</M> that takes <M>b_i</M> to <M>p</M>.
(Note that such a list has holes in all positions corresponding to points
not contained in <M>O^{(i)}</M>.)
<P/>
This approach however will store many different permutations as coset
representatives which can be a problem if the degree <M>n</M> gets bigger.
Our goal therefore is to store as few different permutations as possible such
that we can still reconstruct each representative in <M>R^{(i)}</M>,
and from them the elements in <M>G</M>.
A <E>factorized inverse transversal</E> <M>T</M> is a list
where <M>T[p]</M> is a generator of <M>G^{(i)}</M> such that
<M>p^{{T[p]}}</M> is a point that lies earlier in <M>O^{(i)}</M> than
<M>p</M> (note that we consider <M>O^{(i)}</M> as a list, not as a set).
If we assume inductively that we know an element <M>r \in G^{(i)}</M>
that takes <M>b_i</M> to <M>p^{{T[p]}}</M>,
then <M>r T[p]^{{-1}}</M> is an element in <M>G^{(i)}</M> that takes
<M>b_i</M> to <M>p</M>.
&GAP; uses such factorized inverse transversals.
<P/>
Another name for a factorized inverse transversal is a <E>Schreier tree</E>.
The vertices of the tree are the points in <M>O^{(i)}</M>,
and the root of the tree is <M>b_i</M>.
The edges are defined as the ordered pairs <M>(p, p^{{T[p]}})</M>,
for <M>p \in O^{(i)} \setminus \{ b_i \}</M>. The edge <M>(p, p^{{T[p]}})</M>
is labelled with the generator <M>T[p]</M>, and the product of edge labels
along the unique path from <M>p</M> to <M>b_i</M> is the inverse of the
transversal element carrying <M>b_i</M> to <M>p</M>.
<P/>
Before we describe the construction of stabilizer chains
in&nbsp;<Ref Sect="Construction of Stabilizer Chains"/>,
we explain in&nbsp;<Ref Sect="Randomized Methods for Permutation Groups"/>
the idea of using non-deterministic algorithms;
this is necessary for understanding the options available for the
construction of stabilizer chains.
After that, in&nbsp;<Ref Sect="Stabilizer Chain Records"/> it is explained
how a stabilizer chain is stored in &GAP;,
<Ref Sect="Operations for Stabilizer Chains"/> lists operations for stabilizer
chains,
and&nbsp;<Ref Sect="Low Level Routines to Modify and Create Stabilizer Chains"/>
lists low level routines for manipulating stabilizer chains.

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Randomized Methods for Permutation Groups">
<Heading>Randomized Methods for Permutation Groups</Heading>

<Index Subkey="random">Schreier-Sims</Index>
For most computations with permutation groups,
it is crucial to construct stabilizer chains efficiently.
Sims's original construction in <Cite Key="Sim70"/> is deterministic,
and is called the Schreier-Sims algorithm,
because it is based on Schreier's Lemma (<Cite Key="Hall" Where="p. 96"/>):
given <M>K = \langle S \rangle</M>
and a transversal <M>T</M> for <M>K</M> mod <M>L</M>, one can obtain
<M>|S||T|</M> generators for <M>L</M>. This lemma is applied recursively,
with consecutive point stabilizers <M>G^{(i)}</M> and <M>G^{(i+1)}</M>
playing the role of <M>K</M> and <M>L</M>.
<P/>
In permutation groups of large degree, the number of Schreier generators
to be processed becomes too large, and the deterministic Schreier-Sims
algorithm becomes impractical. Therefore, &GAP; uses randomized
algorithms. The method selection process, which is quite different
from Version 3, works the following way.
<P/>
If a group acts on not more than a hundred points,
Sims's original deterministic algorithm is applied. In groups of
degree greater than hundred, a heuristic algorithm based
on ideas in <Cite Key="BCFS91"/> constructs a stabilizer chain.
This construction is complemented by a
verify-routine that either proves the correctness of the stabilizer chain
or causes the extension of the chain to a correct one.
The user can influence the verification process by setting
the value of the record component <C>random</C>
(cf.&nbsp;<Ref Sect="Construction of Stabilizer Chains"/>).
<P/>
If the <C>random</C> value equals <M>1000</M> then a slight extension of an
unpublished method of Sims is used.
The outcome of this verification process is always correct.
The user also can prescribe any integer <M>x</M>, <M>1 \leq x \leq 999</M>
as the value of <C>random</C>. In this case, a randomized verification
process from <Cite Key="BCFS91"/> is applied, and the result of the
stabilizer chain construction is guaranteed to be correct with probability
at least <M>x/1000</M>. The practical performance of the algorithm is
much better than the theoretical guarantee.
<P/>
If the stabilizer chain is not correct then the elements in the
product of transversals <M>R^{(m)} R^{(m-1)} \cdots R^{(1)}</M> constitute a
proper subset of the group <M>G</M> in question.
This means that a membership test with this stabilizer chain
returns <K>false</K> for
all elements that are not in <M>G</M>,
but it may also return <K>false</K> for some elements of <M>G</M>;
in other words, the result <K>true</K> of a membership test is always correct,
whereas the result <K>false</K> may be incorrect.
<P/>
The construction and verification phases are separated because
there are situations where the verification step can be omitted;
if one happens to know the order of the group in advance then the
randomized construction of the stabilizer chain stops
as soon as the product of the lengths of the basic orbits
of the chain equals the group order, and the chain will be correct
(see the <C>size</C> option of the
<Ref Func="StabChain" Label="for a group (and a record)"/> command).
<P/>
Although the worst case running time is roughly quadratic for
Sims's verification and roughly linear for the randomized one,
in most examples the running time of the stabilizer chain
construction with <C>random</C> value <M>1000</M>
(i.e., guaranteed correct output)
is about the same as the running time of randomized verification
with guarantee of at least <M>90</M> percent correctness.
Therefore, we suggest to use the default value <C>random</C> <M>= 1000</M>.
Possible uses of <C>random</C> values less than <M>1000</M> are
when one has to run through a large collection
of subgroups, and a low value of random is used to choose quickly
a candidate for more thorough examination; another use is when
the user suspects that the quadratic bottleneck of the guaranteed
correct verification is hit.
<P/>
We will give two examples to illustrate these ideas.
<P/>
<Example><![CDATA[
gap> h:= SL(4,7);;
gap> o:= Orbit( h, [1,0,0,0]*Z(7)^0, OnLines );;
gap> op:= Action( h, o, OnLines );;
gap> NrMovedPoints( op );
400
]]></Example>
<P/>
We created a permutation group on <M>400</M> points.
First we compute a guaranteed correct stabilizer chain
(see&nbsp;<Ref Func="StabChain" Label="for a group (and a record)"/>).
<P/>
<Log><![CDATA[
gap> h:= Group( GeneratorsOfGroup( op ) );;
gap> StabChain( h );;  time;
1120
gap> Size( h );
2317591180800
]]></Log>
<P/>
Now randomized verification will be used.
We require that the result is guaranteed correct with probability
<M>90</M> percent.
This means that if we would do this calculation many times over,
&GAP; would <E>guarantee</E> that in least <M>90</M> percent of all
calculations the result is correct.
In fact the results are much better than the guarantee,
but we cannot promise that this will really happen.
(For the meaning of the <C>random</C> component in the second argument of
<Ref Func="StabChain" Label="for a group (and a record)"/>.)
<P/>
First the group is created anew.
<P/>
<Log><![CDATA[
gap> h:= Group( GeneratorsOfGroup( op ) );;
gap> StabChain( h, rec( random:= 900 ) );;  time;
1410
gap> Size( h );
2317591180800
]]></Log>
<P/>
The result is still correct, and the running time is actually somewhat
slower.
If you give the algorithm the order of the group, then it can check
its result, and so things become faster and the result is guaranteed
to be correct. This can be done with the <C>size</C> option
(see&nbsp;<Ref Func="StabChain" Label="for a group (and a record)"/>),
or by setting the size of the group beforehand with <C>SetSize</C>.
<P/>
<Log><![CDATA[
gap> h:=Group( GeneratorsOfGroup( op ) );;
gap> SetSize( h, 2317591180800 );
gap> StabChain( h );;  time;
170
]]></Log>
<P/>
The second example gives a typical group when the verification
with <C>random</C> value <M>1000</M> is slow. The problem is that the group
has a stabilizer subgroup <M>G^{(i)}</M> such that the fundamental
orbit <M>O^{(i)}</M> is split into a lot of orbits when we stabilize
<M>b_i</M> and one additional point of <M>O^{(i)}</M>.
<P/>
<Log><![CDATA[
gap> p1:=PermList(Concatenation([401],[1..400]));;
gap> p2:=PermList(List([1..400],i->(i*20 mod 401)));;
gap> d:=DirectProduct(Group(p1,p2),SymmetricGroup(5));;
gap> h:=Group(GeneratorsOfGroup(d));;
gap> StabChain(h);;time;Size(h);
1030
192480
gap> h:=Group(GeneratorsOfGroup(d));;
gap> StabChain(h,rec(random:=900));;time;Size(h);
570
192480
]]></Log>
<P/>
When stabilizer chains of a group <M>G</M> are created
with <C>random</C> value less than <M>1000</M>,
this is noted in the group <M>G</M>,
by setting of the record component <C>random</C>
in the value of the attribute <Ref Attr="StabChainOptions"/>
for <M>G</M>.
As errors induced by the random methods might propagate,
any group or homomorphism created from <M>G</M> inherits a <C>random</C>
component in its <Ref Attr="StabChainOptions"/> value from the corresponding
component for <M>G</M>.
<P/>
A lot of algorithms dealing with permutation groups use randomized
methods; however, if the initial stabilizer chain construction
for a group is correct, these further methods will provide
guaranteed correct output.

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Construction of Stabilizer Chains">
<Heading>Construction of Stabilizer Chains</Heading>

<#Include Label="StabChain">
<#Include Label="StabChainOptions">
<#Include Label="DefaultStabChainOptions">
<#Include Label="StabChainBaseStrongGenerators">
<#Include Label="MinimalStabChain">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Stabilizer Chain Records">
<Heading>Stabilizer Chain Records</Heading>

If a permutation group has a stabilizer chain,
this is stored as a recursive structure.
This structure is itself a record&nbsp;<A>S</A> and it has
<List>
<Mark>(1)</Mark>
<Item>
  components that provide information about one level&nbsp;<M>G^{(i)}</M>
  of the stabilizer chain (which we call the <Q>current stabilizer</Q>) and
</Item>
<Mark>(2)</Mark>
<Item>
  a component <C>stabilizer</C> that holds another such record, namely the
  stabilizer chain of the next stabilizer&nbsp;<M>G^{(i+1)}</M>.
</Item>
</List>
<P/>
This gives a recursive structure where the <Q>outermost</Q> record
representing the <Q>topmost</Q> stabilizer is bound to
the group record component <C>stabChain</C> and has the components
explained below.
Note: Since the structure is recursive, <E>never print a stabilizer chain!</E>
(Unless you want to exercise the scrolling capabilities of your terminal.)
<List>
<Mark><C>identity</C> </Mark>
<Item>
        the identity element of the current stabilizer.
</Item>
<Mark><C>labels</C> </Mark>
<Item>
        a list of permutations which contains labels for the Schreier tree
        of  the current stabilizer,  i.e., it contains
        elements for the factorized
        inverse transversal.  The  first entry  in this list  is
        always the  <C>identity</C>.
        Note that &GAP; tries to arrange things so that the <C>labels</C>
        components are identical (i.e., the same &GAP; object)
        in every stabilizer of the chain;
        thus the <C>labels</C> of a stabilizer do not necessarily all lie
        in the this stabilizer (but see <C>genlabels</C> below).
</Item>
<Mark><C>genlabels</C> </Mark>
<Item>
        a list of integers indexing some of the permutations in the
        <C>labels</C> component. The    <C>labels</C> addressed in this   way
        form a generating set for the current stabilizer. If the
        <C>genlabels</C> component is empty,  the rest of the stabilizer chain
        represents  the trivial subgroup, and  can be ignored, e.g., when
        calculating the size.
</Item>
<Mark><C>generators</C> </Mark>
<Item>
        a list of generators for the current stabilizer. Usually, it is
        <C>labels{ genlabels }</C>.
</Item>
<Mark><C>orbit</C> </Mark>
<Item>
        the vertices of  the Schreier tree,  which  form the basic  orbit
        <M>b_i^{{G^{(i)}}}</M>, ordered in such a way that the base point
        <M>b_i</M> is in the first position in the orbit.
</Item>
<Mark><C>transversal</C></Mark>
<Item>
  The factorized inverse transversal found during the orbit algorithm.
  The element <M>g</M> stored at <C>transversal</C><M>[i]</M> will map
  <M>i</M> to another point <M>j</M> that in the Schreier tree is closer to
  the base point.
  By iterated application (<C>transversal</C><M>[j]</M> and so on) eventually
  the base point is reached and an element that maps <M>i</M> to the base
  point found as product.
</Item>
<Mark><C>translabels</C></Mark>
<Item>
  An index list such that
  <C>transversal</C><M>[j] =</M>
  <C>labels</C><M>[</M> <C>translabels</C><M>[j] ]</M>.
  This list takes up comparatively little memory and is used to speed
  up base changes.
</Item>
<Mark><C>stabilizer</C> </Mark>
<Item>
  If the current stabilizer is not yet the trivial group,
  the stabilizer chain continues with the stabilizer of the current
  base point, which is again represented as a record with components
  <C>labels</C>, <C>identity</C>, <C>genlabels</C>, <C>generators</C>,
  <C>orbit</C>, <C>translabels</C>, <C>transversal</C> (and perhaps
  <C>stabilizer</C>).
  This record is bound to the <C>stabilizer</C> component of the current
  stabilizer.
  The last member of a stabilizer chain is recognized by the fact that it has
  no <C>stabilizer</C> component bound.
</Item>
</List>
It is possible that different stabilizer  chains share the same record as
one of their iterated <C>stabilizer</C> components.
<P/>
<Example><![CDATA[
gap> g:=Group((1,2,3,4),(1,2));;
gap> StabChain(g);
<stabilizer chain record, Base [ 1, 2, 3 ], Orbit length 4, Size: 24>
gap> BaseOfGroup(g);
[ 1, 2, 3 ]
gap> StabChainOptions(g);
rec( random := 1000 )
gap> DefaultStabChainOptions;
rec( random := 1000, reduced := true, tryPcgs := true )
]]></Example>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Operations for Stabilizer Chains">
<Heading>Operations for Stabilizer Chains</Heading>

<#Include Label="BaseStabChain">
<#Include Label="BaseOfGroup">
<#Include Label="SizeStabChain">
<#Include Label="StrongGeneratorsStabChain">
<#Include Label="GroupStabChain">
<#Include Label="OrbitStabChain">
<#Include Label="IndicesStabChain">
<#Include Label="ListStabChain">
<#Include Label="ElementsStabChain">
<#Include Label="IteratorStabChain">
<#Include Label="InverseRepresentative">
<#Include Label="SiftedPermutation">
<#Include Label="MinimalElementCosetStabChain">
<#Include Label="LargestElementStabChain">
<#Include Label="ApproximateSuborbitsStabilizerPermGroup">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Low Level Routines to Modify and Create Stabilizer Chains">
<Heading>Low Level Routines to Modify and Create Stabilizer Chains</Heading>

These operations modify a stabilizer chain or obtain new chains with
specific properties. They are rather technical and should only be used if
such low-level routines are deliberately required. (For all functions in
this section the parameter <A>S</A> is a stabilizer chain.)

<#Include Label="CopyStabChain">
<#Include Label="CopyOptionsDefaults">
<#Include Label="ChangeStabChain">
<#Include Label="ExtendStabChain">
<#Include Label="ReduceStabChain">
<!-- %%declaration{ConjugateStabChain} -->
<#Include Label="RemoveStabChain">
<#Include Label="EmptyStabChain">
<#Include Label="InsertTrivialStabilizer">
<#Include Label="IsFixedStabilizer">
<#Include Label="AddGeneratorsExtendSchreierTree">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Backtrack">
<Heading>Backtrack</Heading>

A main use for stabilizer chains is in backtrack algorithms for permutation
groups. &GAP; implements a partition-backtrack algorithm as described in
<Cite Key="Leon91"/> and refined in <Cite Key="Theissen97"/>.
<P/>
<#Include Label="SubgroupProperty">
<#Include Label="ElementProperty">
<#Include Label="TwoClosure">
<#Include Label="InfoBckt">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Working with large degree permutation groups">
<Heading>Working with large degree permutation groups</Heading>

Permutation groups of large degree (usually at least a few <M>10000</M>)
can pose a challenge to the heuristics used in the algorithms for
permutation groups.
This section lists a few useful tricks that may speed up calculations with
such large groups enormously.
<P/>
The first aspect concerns solvable groups: A lot of calculations (including
an initial stabilizer chain computation thanks to the algorithm from
<Cite Key="Sims90b"/>) are faster if a permutation group is known to be
solvable.
On the other hand, proving nonsolvability can be expensive for higher
degrees. Therefore &GAP; will automatically test a permutation group for
solvability, only if the degree is not exceeding <M>100</M>.
(See also the <C>tryPcgs</C> component of <Ref Attr="StabChainOptions"/>.)
It is therefore beneficial to tell a group of larger degree, which is known
to be solvable, that it is, using <C>SetIsSolvableGroup(<A>G</A>,true)</C>.
<P/>
The second aspect concerns memory usage.
A permutation on more than <M>65536</M> points requires <M>4</M> bytes per
point for storing.
So permutations on <M>256000</M> points require roughly 1MB of storage
per permutation. Just storing the
permutations required for a stabilizer chain might already go beyond the
available memory, in particular if the base is not very short. In such a
situation it can be useful, to replace the permutations by straight line
program elements (see&nbsp;<Ref Sect="Straight Line Program Elements"/>).
<P/>
The following code gives an example of usage:
We create a group of degree <M>231000</M>.
Using straight line program elements,
one can compute a stabilizer chain in about <M>200</M> MB of memory.
<P/>
<!-- % don't attempt to run this example through the manual checker - it takes -->
<!-- % <E>ages</E> to run and <E>lots</E> of memory! -->
<!-- % besides the input data is not given ... -->
<Log><![CDATA[
gap> Read("largeperms"); # read generators from file
gap> gens:=StraightLineProgGens(permutationlist);;
gap> g:=Group(gens);
<permutation group with 5 generators>
gap> # use random algorithm (faster, but result is monte carlo)
gap> StabChainOptions(g).random:=1;;
gap> Size(g); # enforce computation of a stabilizer chain
3529698298145066075557232833758234188056080273649172207877011796336000
]]></Log>
<P/>
Without straight line program elements, the same calculation runs into
memory problems after a while even with 512MB of workspace:
<Log><![CDATA[
gap> h:=Group(permutationlist);
<permutation group with 5 generators>
gap> StabChainOptions(h).random:=1;;
gap> Size(h);
exceeded the permitted memory (`-o' command line option)
...
]]></Log>
<P/>
The advantage in memory usage however is paid for in runtime: Comparisons of
elements become much more expensive. One can avoid some of the related
problems by registering a known base with the straight line program elements
(see&nbsp;<Ref Func="StraightLineProgGens"/>).
In this case element comparison will only
compare the images of the given base points.
If we are planning to do extensive calculations with the group, it can even
be worth to recreate it with straight line program elements knowing a
previously computed base:
<P/>
<Log><![CDATA[
gap> # get the base we computed already
gap> bas:=BaseStabChain(StabChainMutable(g));
[ 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55,
...
  2530, 2533, 2554, 2563, 2569 ]
gap> gens:=StraightLineProgGens(permutationlist,bas);;
gap> g:=Group(gens);;
gap> SetSize(g,
> 3529698298145066075557232833758234188056080273649172207877011796336000);
gap> Random(g);; # enforce computation of a stabilizer chain
]]></Log>
<P/>
As we know already base and size, this second stabilizer chain calculation
is much faster than the first one and takes less memory.

</Section>
</Chapter>


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