%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%A  grpperm.msk                 GAP documentation            Alexander Hulpke
%A                                                           Heiko Theissen
%A                                                           Akos Seress
%%
%A  @(#)$Id: grpperm.msk,v 1.39.2.1 2007/08/28 12:58:51 gap Exp $
%%
%Y  (C) 1998 School Math and Comp. Sci., University of St.  Andrews, Scotland
%Y  Copyright (C) 2002 The GAP Group
%%
\Chapter{Permutation Groups}

\Declaration{IsPermGroup}
\beginexample
gap> g:=Group((1,2,3,4),(1,2));
Group([ (1,2,3,4), (1,2) ])
\endexample

Permutation groups are groups and therefore all operations for groups (see
Chapter~"Groups") can be applied to them. In many cases special methods are
installed for permutation groups that make computations more effective.

%%  The code for the computation and verification stabilizer chains, for
%%  composition series and the radical is due to \'Akos Seress.
%%  Heiko Thei{\ss}en wrote the backtrack routines which are used to compute for
%%  example centralizers, normalizers and conjugating elements.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{The Natural Action}

The functions `MovedPoints', `NrMovedPoints', `LargestMovedPoint',
and `SmallestMovedPoint' are defined for arbitrary collections of
permutations (see~"Moved Points of Permutations"),
in particular they can be applied to permutation groups.
\beginexample
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
\endexample

The action of a permutation group on the positive integers is a group
action (via the acting function `OnPoints').
Therefore all action functions can be applied
(see the Chapter~"Group Actions"),
for example `Orbit', `Stabilizer', `Blocks', `IsTransitive', `IsPrimitive'.

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 `^', the following functions are provided for this purpose.

\Declaration{OrbitPerms}
\Declaration{OrbitsPerms}
\beginexample
gap> OrbitPerms( [ (1,2,3)(4,5), (3,6) ], 1 );
[ 1, 2, 3, 6 ]
gap> OrbitsPerms( [ (1,2,3)(4,5), (3,6) ], [ 1 .. 6 ] );
[ [ 1, 2, 3, 6 ], [ 4, 5 ] ]
\endexample

Similarly, several functions concerning the natural action of
permutation groups address stabilizer chains (see~"Stabilizer Chains")
rather than permutation groups themselves, for example `BaseStabChain'
(see~"BaseStabChain").


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Computing a Permutation Representation}

\Declaration{IsomorphismPermGroup}
\beginexample
gap> g:=SmallGroup(24,12);
<pc group of size 24 with 4 generators>
gap> iso:=IsomorphismPermGroup(g);
<action isomorphism>
gap> Image(iso,g.3*g.4);
(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)
\endexample

In many cases the permutation representation constructed by
`IsomorphismPermGroup' is regular.

\Declaration{SmallerDegreePermutationRepresentation}
\beginexample
gap> image:= Image( iso );;  NrMovedPoints( image );
24
gap> small:= SmallerDegreePermutationRepresentation( image );;
gap> Image( small );
Group([ (2,3), (2,4,3), (1,3)(2,4), (1,2)(3,4) ])
\endexample

% 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> Length(MovedPoints(Image(q)));
% 6


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Symmetric and Alternating Groups}

The commands `SymmetricGroup' and `AlternatingGroup' (see~"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 *natural* symmetric or alternating group,
respectively.

\Declaration{IsNaturalSymmetricGroup}
\Declaration{IsNaturalAlternatingGroup}

For groups that are known to be natural symmetric or natural alternating
groups, very efficient methods for computing membership, conjugacy classes,
Sylow subgroups etc.~are used.
\beginexample
gap> g:=Group((1,5,7,8,99),(1,99,13,72));;
gap> IsNaturalSymmetricGroup(g);
true
gap> g;
Sym( [ 1, 5, 7, 8, 13, 72, 99 ] )
gap> IsNaturalSymmetricGroup( Group( (1,2)(4,5), (1,2,3)(4,5,6) ) );
false
\endexample

The following functions can be used to check whether a given group
(not necessarily a permutation group)
is isomorphic to a symmetric or alternating group.

There are no methods yet for IsSymmetricGroup and IsAlternatingGroup!

\Declaration{IsSymmetricGroup}
\Declaration{IsAlternatingGroup}


\Declaration{SymmetricParentGroup}
\beginexample
gap> SymmetricParentGroup( Group( (1,2), (4,5), (7,8,9) ) );
Sym( [ 1, 2, 4, 5, 7, 8, 9 ] )
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Primitive Groups}

\Declaration{ONanScottType}

\Declaration{SocleTypePrimitiveGroup}

\beginexample
gap> g:=AlternatingGroup(5);;
gap> h:=DirectProduct(g,g);;
gap> p:=List([1,2],i->Projection(h,i));;
gap> ac:=Action(h,AsList(g),
> function(g,h) return Image(p[1],h)^-1*g*Image(p[2],h);end);;
gap> Size(ac);NrMovedPoints(ac);IsPrimitive(ac,[1..60]);
3600
60
true
gap> ONanScottType(ac);
"3a"
gap> SocleTypePrimitiveGroup(ac);
rec( series := "A", width := 2, 
  name := "A(5) ~ A(1,4) = L(2,4) ~ B(1,4) = O(3,4) ~ C(1,4) = S(2,4) ~ 2A(1,4\
) = U(2,4) ~ A(1,5) = L(2,5) ~ B(1,5) = O(3,5) ~ C(1,5) = S(2,5) ~ 2A(1,5) = U\
(2,5)", parameter := 5 )
\endexample


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Stabilizer Chains}

Many of the algorithms for permutation groups use a *stabilizer chain* of
the group.
The concepts of stabilizer chains, *bases*, and *strong generating sets* were
introduced by Charles Sims in~\cite{Sim70}.
A further discussion of base change is given in
section~"ext:Generalized Conjugation Technique" in ``Extending {\GAP}''.

Let $B=[b_1, \ldots, b_n]$ be a list of points, $G^{(1)} = G$ and $G^{(i+1)} =
Stab_{G^{(i)}}(b_i)$, such that $G^{(n+1)} = \{ () \}$.
Then the list $[b_1, \ldots, b_n]$ is called a *base* of $G$,
the points $b_i$ are called *base
points*.  A set $S$ of generators for $G$ satisfying the condition $\< S
\cap G^{(i)} >$ = $G^{(i)}$ for each $1 \leq i \leq n$, is called a *strong
generating set* (SGS) of $G$. (More precisely we ought to say that it is a
SGS of $G$ *relative* to $B$).
The chain of subgroups $G^{(i)}$ of $G$ itself is
called the *stabilizer chain* of $G$ relative to $B$.

Since $[b_1, \ldots, b_n]$, where $n$ is the degree of $G$ and $b_i$ are the
moved points of $G$, certainly is a base for $G$ there exists a base for
each permutation group. The number of points in a base is called the
*length* of the base. A base $B$ is called *reduced* if there exists no $i$
such that $G^{(i)} = G^{(i+1)}$. (This however does not imply that no subset
of $B$ could also serve as a base.)
Note that different reduced bases for one permutation group $G$
may have different lengths.
For example, the irreducible degree $416$ permutation representation
of the Chevalley Group $G_2(4)$ possesses reduced bases of length 5 and 7.

Let $R^{(i)}$ be a right transversal of $G^{(i+1)}$ in $G^{(i)}$, i.e. a set
of right coset representatives of the cosets of $G^{(i+1)}$ in $G^{(i)}$.
Then each element $g$ of $G$ has a unique representation of the form $g =
r_n \ldots r_1$ with $r_i \in R^{(i)}$.
The cosets of $G^{(i+1)}$ in $G^{(i)}$ are in bijective correspondence with
the points in $O^{(i)} := b_i^{G^{(i)}}$.
So we could represent a transversal as a list $T$ such that $T[p]$ is a
representative of the coset corresponding to the point $p \in O^{(i)}$,
i.e., an element of $G^{(i)}$ that takes $b_i$ to $p$.
(Note that such a list has holes in all positions corresponding to points
not contained in $O^{(i)}$.)

This approach however will store many different permutations as coset
representatives which can be a problem if the degree $n$ gets bigger. Our
goal therefore is to store as few different permutations as possible such
that we can still reconstruct each representative in $R^{(i)}$, and from
them the elements in $G$. A *factorized inverse transversal* $T$ is a list
where $T[p]$ is a generator of $G^{(i)}$ such that $p^{T[p]}$ is a point
that lies earlier in $O^{(i)}$ than $p$ (note that we consider $O^{(i)}$ as
a list, not as a set). If we assume inductively that we know an element $r
\in G^{(i)}$ that takes $b_i$ to $p^{T[p]}$, then $r T[p]^{-1}$ is an
element in $G^{(i)}$ that takes $b_i$ to $p$. {\GAP} uses such factorized
inverse transversals.

Another name for a factorized inverse transversal is a
*Schreier tree*. The vertices of the tree are the points in $O^{(i)}$,
and the root of the tree is $b_i$.
The edges are defined as the ordered pairs $(p, p^{T[p]})$,
for $p \in O^{(i)} \setminus \{ b_i\}$. The edge $(p, p^{T[p]})$
is labelled with the generator $T[p]$, and the product of edge labels
along the unique path from $p$ to $b_i$ is the inverse of the
transversal element carrying $b_i$ to $p$.

Before we describe the construction of stablizer chains
in~"Construction of Stabilizer Chains",
we explain in~"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~"Stabilizer Chain Records" it is explained
how a stabilizer chain is stored in {\GAP},
"Operations for Stabilizer Chains" lists operations for stabilizer
chains,
and~"Low Level Routines to Modify and Create Stabilizer Chains" lists
low level routines for manipulating stabilizer chains.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Randomized Methods for Permutation Groups}

\atindex{Schreier-Sims!random}{@Schreier-Sims!random}
For most computations with permutation groups,
it is crucial to construct stabilizer chains efficiently. Sims's
original construction \cite{Sim70} is deterministic, and is called
the Schreier-Sims algorithm, because it is based
on Schreier's Lemma (p. 96 in \cite{Hall}): given $K=\langle S \rangle$
and a transversal $T$ for $K$ mod $L$, one can obtain
$|S||T|$ generators for $L$. This lemma is applied recursively,
with consecutive point stabilizers $G^{(i)}$ and $G^{(i+1)}$
playing the role of $K$ and $L$.

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.

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{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 `random'
(cf.~"Construction of Stabilizer Chains").

If `random' $=1000$ 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 $1 \le x \le 999$
as the value of `random'. In this case, a randomized verification
process from \cite{BCFS91} is applied, and the result of the
stabilizer chain construction is guaranteed to be correct with probability
at least $x/1000$. The practical performance of the algorithm is
much better than the theoretical guarantee.

If the stabilizer chain is not correct then the elements in the
product of transversals $R^{(m)}R^{(m-1)}\cdots R^{(1)}$ constitute a
proper subset of the group $G$ in question.
This means that a membership test with this stabilizer chain
returns `false' for
all elements that are not in $G$,
but it may also return `false' for some elements of $G$;
in other words, the result `true' of a membership test is always correct,
whereas the result `false' may be incorrect.

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 `size' option of the `StabChain' command in~"StabChain").

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 `random'$=1000$ (i.e., guaranteed correct output)
is about the same as the running time of randomized verification
with guarantee of at least $90\%$ correctness. Therefore, we suggest
to use the default value `random'$=1000$. Possible uses of
`random'$\<1000$ 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.

We will illustrate these ideas in two examples.

\beginexample
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
\endexample

We created a permutation group on 400 points.
First we compute a guaranteed correct stabilizer chain.
(The `StabChain' command is described in~"StabChain".)

%notest
\beginexample
gap> h:= Group( GeneratorsOfGroup( op ) );;
gap> StabChain( h );;  time;
1120
gap> Size( h );
2317591180800
\endexample

Now randomized verification will be used.
We require that the result is guaranteed correct with probability $90\%$.
This means that if we would do this calculation many times over,
{\GAP} would *guarantee* that in least $90\%$ 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 `random' component in the second argument of
`StabChain', see~"StabChain".)

First the group is created anew.

%notest
\beginexample
gap> h:= Group( GeneratorsOfGroup( op ) );;
gap> StabChain( h, rec( random:= 900 ) );;  time;
1410
gap> Size( h );
2317591180800
\endexample

The result is still correct, and the running time is actually somewhat
slower.
If you give the algorithm additional information so that it can check
its results,
things become faster and the result is guaranteed to be correct.

%notest
\beginexample
gap> h:=Group( GeneratorsOfGroup( op ) );;
gap> SetSize( h, 2317591180800 );
gap> StabChain( h );;  time;
170
\endexample

The second example gives a typical group when the verification
with `random' $=1000$ is slow. The problem is that the group
has a stabilizer subgroup $G^{(i)}$ such that the fundamental
orbit $O^{(i)}$ is split into a lot of orbits when we stabilize
$b_i$ and one additional point of $O^{(i)}$.

%notest
\beginexample
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
\endexample

When stabilizer chains of a group $G$ are created
with `random' $\<1000$, this is noted in the group $G$,
by setting of the record component `random'
in the value of the attribute `StabChainOptions' for $G$
(see~"StabChainOptions").
As errors induced by the random methods might propagate,
any group or homomorphism created from $G$ inherits a `random' component
in its `StabChainOptions' from the corresponding component for $G$.

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{Construction of Stabilizer Chains}

\Declaration{StabChain}

\Declaration{StabChainOptions}
\Declaration{DefaultStabChainOptions}
\Declaration{StabChainBaseStrongGenerators}

\Declaration{MinimalStabChain}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Stabilizer Chain Records}

If a  permutation group  has a   stabilizer chain, this   is stored  as a
recursive structure. This  structure  is itself  a record~<S>  and it has
(1)~components that provide information about  one level~$G^{(i)}$ of the
stabilizer chain (which  we call  the  ``current stabilizer'')  and (2)~a
component `stabilizer'  that  holds   another such  record,   namely  the
stabilizer chain  of    the next stabilizer~$G^{(i+1)}$.  This  gives   a
recursive  structure  where the  ``outermost''  record  representing  the
``topmost'' stabilizer is bound to the group record component `stabChain'
and   has the components explained below.
Note: Since the structure is recursive, *never print a stabilizer chain!*
(Unless you want to exercise the scrolling capabilities of your terminal.)
\beginitems
`identity' &
        the identity element of the current stabilizer.

`labels' &
        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  is this list  is
        always the  `identity'.
        Note that {\GAP} tries to arrange things so that the `labels'
        components are identical (i.e., the same {\GAP} object)
        in every stabilizer of the chain;
        thus the `labels' of a stabilizer do not necessarily all lie
        in the this stabilizer (but see `genlabels' below).

`genlabels' &
        a list of integers indexing some of the permutations in the
        `labels' component. The    `labels' addressed in this   way
        form a generating set for the current stabilizer. If the
        `genlabels' component is empty,  the rest of the stabilizer chain
        represents  the trivial subgroup, and  can be ignored, e.g., when
        calculating the size.

`generators' &
        a list of generators for the current stabilizer. Usually, it is
        `labels\{ genlabels \}'.

`orbit' &
        the vertices of  the Schreier tree,  which  form the basic  orbit
        $b_i^{G^{(i)}}$, ordered in such a way that the  base point $b_i$ is

`transversal' &
	The factorized inverse transversal found during the orbit algorithm.
	The element <g> stored at `transversal[<i>]' will map <i> to another
	point <j> that in the Schreier tree is closer to the base point. By
	iterated application (`transversal[<j>]' and so on) eventually the
	base point is reached and an element that maps <i> to the base
	point foiund as product.

`translabels' &
	An index list such that 
	`transversal[<j>] =  labels[ translabels[<j>] ]'.
	This list takes up comparatively little memory and is used to speed
	up base changes.

`stabilizer' &
        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 `labels',    `identity',    `genlabels', `generators',
        `orbit',     `translabels',  `transversal'     (and      possibly
        `stabilizer'). This record is bound to the `stabilizer' component
        of the current stabilizer. The last member  of a stabilizer chain
        is recognized  by the fact that  it has no `stabilizer' component
        bound.
\enditems
It is possible that different stabilizer  chains share the same record as
one of their iterated `stabilizer' components.

\beginexample
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( reduced := true, random := 1000, tryPcgs := true )
\endexample


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Operations for Stabilizer Chains}

\Declaration{BaseStabChain}
\Declaration{BaseOfGroup}

\Declaration{SizeStabChain}
\Declaration{StrongGeneratorsStabChain}
\Declaration{GroupStabChain}
\Declaration{OrbitStabChain}
\Declaration{IndicesStabChain}
\Declaration{ListStabChain}
\Declaration{ElementsStabChain}

\Declaration{InverseRepresentative}
\Declaration{SiftedPermutation}
\Declaration{MinimalElementCosetStabChain}
\Declaration{LargestElementStabChain}

\Declaration{ApproximateSuborbitsStabilizerPermGroup}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Low Level Routines to Modify and Create Stabilizer Chains}

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 <S> is a stabilizer chain.)

\Declaration{CopyStabChain}
\Declaration{CopyOptionsDefaults}
\Declaration{ChangeStabChain}
\Declaration{ExtendStabChain}
\Declaration{ReduceStabChain}
%%declaration{ConjugateStabChain}
\Declaration{RemoveStabChain}
\Declaration{EmptyStabChain}
\Declaration{InsertTrivialStabilizer}
\Declaration{IsFixedStabilizer}
\Declaration{AddGeneratorsExtendSchreierTree}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Backtrack}

A main use for stabilizer chains is in backtrack algorithms for permutation
groups. {\GAP} implements a partition-backtrack algorithm as described in
\cite{Leon91} and refined in \cite{Theissen97}.

\Declaration{SubgroupProperty}
\Declaration{ElementProperty}

A typical example of using the optional subgroups <L> and <R> is the
conjugacy test for elements <a> and <b> for which one can set
$<L>:=C_G(<a>)$ and $<R>:=C_G(<b>)$.

\beginexample
gap> propfun:= el -> (1,2,3)^el in [ (1,2,3), (1,3,2) ];;
gap> SubgroupProperty( g, propfun, Subgroup( g, [ (1,2,3) ] ) );
Group([ (1,2,3), (2,3) ])
gap> ElementProperty( g, el -> Order( el ) = 2 );
(2,4)
\endexample

Chapter~"Permutations" describes special operations to construct
permutations in the symmetric group without using backtrack constructions.

Backtrack routines are also called by the methods for permutation groups
that compute centralizers, normalizers, intersections, conjugating elements
as well as
stabilizers for the operations of a permutation group
`OnPoints', `OnSets', `OnTuples' and `OnSetSets'. Some of these methods use
more specific refinements than `SubgroupProperty' or `ElementProperty'.
For the definition of refinements, and how one can define refinements, see
Section~"ext:The General Backtrack Algorithm with Ordered Partitions" in
``Extending {\GAP}''.

\Declaration{TwoClosure}
\beginexample
gap> TwoClosure(Group((1,2,3),(2,3,4)));
Sym( [ 1 .. 4 ] )
\endexample

\Declaration{InfoBckt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Working with large degree permutation groups}

Permutation groups of large degree (usually at least a few 10000) 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.

The first aspect concerns solvable groups: A lot of calculations (including
an initial stabilizer chain computation thanks to the algorithm from
\cite{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 100.
(See also the `tryPcgs' component of `StabChainOptions'.)
It is therefore beneficial to tell a group of larger degree, which is known
to be solvable, that it is, using `SetIsSolvableGroup(<G>,true)'.

The second aspect concerns memory usage. A permutation on more than 65536
points requires 4 byte per point for storing. So permutations on 256000
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~"Straight Line Program Elements").

The following code gives an example of usage:
We create a group of degree 231000. Using straight line program elements,
one can compute a stabilizer chain in about 200 MB of memory.

% don't attempt to run this example through the manual checker -- it takes
% *ages* to run and *lots* of memory!
% besides the input data is not given ...
\begintt
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
\endtt

Without straight line program elements, the same calculation runs into
memory problems after a while even with 512MB of workspace:
\begintt
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) at
mlimit := 1; called from
SCRMakeStabStrong( S.stabilizer, [ g ], param, orbits, where, basesize,
 base, correct, missing, false ); called from
 SCRMakeStabStrong( S.stabilizer, [ g ], param, orbits, where, basesize,
...
\endtt

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~`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:

\begintt
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
\endtt

As we know already base and size, this second stabilizer chain calculation
is much faster than the first one and takes less memory.

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