<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <!-- %% --> <!-- %W stbchain.tex GAP documentation Heiko Theißen --> <!-- %% --> <!-- %% --> <!-- %Y Copyright 1997, Lehrstuhl D für Mathematik, RWTH Aachen, Germany --> <!-- %% -->
<Chapter Label="More about Stabilizer Chains">
<Heading>More about Stabilizer Chains</Heading>
This chapter contains some rather technical complements to the material
handled in the chapters <Ref Chap="Permutations"/>
and <Ref Chap="Permutation Groups"/>.
The command <C>ConjugateGroup( <A>G</A>, <A>p</A> )</C>
(see <Ref Oper="ConjugateGroup"/>)
for a permutation group <A>G</A> with stabilizer chain
equips its result also with a stabilizer chain, namely with the chain of
<A>G</A> conjugate by <A>p</A>. Conjugating a stabilizer chain by a permutation <A>p</A>
means replacing all the points which appear in the <C>orbit</C> components by
their images under <A>p</A> and replacing every permutation <A>g</A> which appears
in a <C>labels</C> or <C>transversal</C> component by its conjugate <M>g^p</M>. The
conjugate <M>g^p</M> acts on the mapped points exactly as <A>g</A> did on the
original points, i.e., <M>(pnt.p). g^p = (pnt.g).p</M>. Since the entries in
the <C>translabels</C> components are integers pointing to positions of the
<C>labels</C> list, the <C>translabels</C> lists just have to be permuted by <A>p</A>
for the conjugated stabilizer.
Then <C>generators</C> is reconstructed as
<C>labels{ genlabels }</C> and
<C>transversal{ orbit }</C> as
<C>labels{ translabels{ orbit } }</C>.
<P/>
<Index>generalized conjugation technique</Index>
This conjugation technique can be generalized. Instead of mapping points
and permutations under the same permutation <A>p</A>, it is sometimes
desirable (e.g., in the context of permutation group homomorphisms) to
map the points with an arbitrary mapping <M>map</M> and the permutations with
a homomorphism <M>hom</M> such that the compatibility of the actions is still
valid: <M>map(pnt).hom(g) = map(pnt.g)</M>. (Of course the ordinary
conjugation is a special case of this, with <M>map(pnt) = pnt.p</M> and
<M>hom(g) = g^p</M>.)
<P/>
In the generalized case, the <Q>conjugated</Q> chain need not be a
stabilizer chain for the image of <M>hom</M>, since the <Q>preimage</Q> of the
stabilizer of <M>map(b)</M> (where <M>b</M> is a base point) need not fix <M>b</M>, but
only fixes the preimage <M>map^{{-1}}( map(b) )</M> setwise. Therefore the method
can be applied only to one level and the next stabilizer must be computed
explicitly.
But if <M>map</M> is injective, we have <M>map(b).hom(g) = map(b)</M> if and
only if <M>b.g = b</M>, and if this holds, then <M>g = w(g_1, \ldots, g_n)</M>
is a word in the
generators <M>g_1, \ldots, g_n</M> of the stabilizer of <M>b</M> and <!-- % replaced \buildrel *\over= by {$*$}{$=$} ... easiest compromise for HTML -->
<M>hom(g) =^* w( hom(g_1), \ldots, hom(g_n) )</M> is in the
<Q>conjugated</Q> stabilizer.
If, more generally, <M>hom</M> is a right inverse to a homomorphism
<M>\varphi</M> (i.e., <M>\varphi(hom(g)) = g</M> for all <M>g</M>),
equality <M>*</M> holds modulo the kernel of <M>\varphi</M>;
in this case the <Q>conjugated</Q> chain can be made into a real stabilizer
chain by extending each level with the generators of the kernel and
appending a proper stabilizer chain of the kernel at the end.
These special cases will occur in the algorithms for permutation group
homomorphisms (see <Ref Chap="Group Homomorphisms"/>).
<P/>
To <Q>conjugate</Q> the points (i.e., <C>orbit</C>) and permutations (i.e.,
<C>labels</C>) of the Schreier tree, a loop is set up over the <C>orbit</C> list
constructed during the orbit algorithm, and for each vertex <M>b</M> with
unique edge <M>a(l)b</M> ending at <M>b</M>, the label <M>l</M> is mapped with <M>hom</M> and
<M>b</M> with <M>map</M>. We assume that the <C>orbit</C> list was built w.r.t. a
certain ordering <M><</M> of the labels,
where <M>l' < l means that every point in
the orbit was mapped with <M>l' before it was mapped with l. This shape
of the <C>orbit</C> list is guaranteed if the Schreier tree is extended only
by <Ref Func="AddGeneratorsExtendSchreierTree"/>,
and it is then also guaranteed for
the <Q>conjugated</Q> Schreier tree. (The ordering of the labels cannot be
read from the Schreier tree, however.)
<P/>
In the generalized case, it can happen that the edge <M>a(l)b</M> bears a
label <M>l</M> whose image is <Q>old</Q>, i.e., equal to the image of an earlier
label <M>l' < l. Because of the compatibility of the actions we then have
<M>map(b) = map(a).hom(l)^{{-1}} = map(a).hom(l')^{{-1}} = map(a{{l'}}^{{-1}})</M>,
so <M>map(b)</M> is already equal to the image of the vertex <M>a{{l'}}^{{-1}}.
This vertex must have been encountered before <M>b = al^{{-1}}</M> because
<M>l' < l. We conclude that the image of a label can be old only if the
vertex at the end of the corresponding edge has an <Q>old</Q> image, too,
but then it need not be <Q>conjugated</Q> at all. A similar remark applies
to labels which map under <M>hom</M> to the identity.
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="The General Backtrack Algorithm with Ordered Partitions">
<Heading>The General Backtrack Algorithm with Ordered Partitions</Heading>
Section <Ref Sect="Backtrack"/> describes the basic
functions for a backtrack search. The purpose of
this section is to document how the general backtrack algorithm is
implemented in &GAP; and which parts you have to modify if you want to
write your own backtrack routines.
<Subsection Label="Internal representation of ordered partitions">
<Heading>Internal representation of ordered partitions</Heading>
<Index Subkey="internal representation">ordered partitions</Index>
&GAP; represents an ordered partition as a record with the following
components.
<List>
<Mark><C>points</C></Mark>
<Item>
a list of all points contained in the partition, such that the
points of each cell from lie consecutively,
</Item>
<Mark><C>cellno</C></Mark>
<Item>
a list whose <A>i</A>th entry is the number of the cell which contains
the point <A>i</A>,
</Item>
<Mark><C>firsts</C></Mark>
<Item>
a list such that <C>points[firsts[<A>j</A>]]</C> is the first point in
<C>points</C> which is in cell <A>j</A>,
</Item>
<Mark><C>lengths</C></Mark>
<Item>
a list of the cell lengths.
</Item>
</List>
Some of the information is redundant, e.g., the <C>lengths</C> could also be
read off the <C>firsts</C> list, but since this need not be increasing, it
would require some searching. Similar for <C>cellno</C>, which could be
replaced by a systematic search of <C>points</C>, keeping track of what cell
is currently being traversed.
With the above components, the <A>m</A>th cell of a partition <A>P</A>
is expressed as <C><A>P</A>.points{ [ <A>P</A>.firsts[<A>m</A>] ..
<A>P</A>.firsts[<A>m</A>] + <A>P</A>.lengths[<A>m</A>] - 1 ] }</C>.
The most important
operations, however, to be performed upon <A>P</A> are the splitting of a cell
and the reuniting of the two parts. Following the strategy of J. Leon, <!-- explicit citation, please -->
this is done as follows:
<P/>
<List>
<Mark>(1)</Mark>
<Item>
The points which make up the cell that is to be split are sorted so that
the ones that remain inside occupy positions <C>[ <A>P</A>.firsts[<A>m</A>] .. <A>last</A>
]</C> in the list <C><A>P</A>.points</C> (for a suitable value of <A>last</A>).
</Item>
<Mark>(2)</Mark>
<Item>
The points at positions <C>[ <A>last</A> + 1 .. <A>P</A>.firsts[<A>m</A>] +
<A>P</A>.lengths[<A>m</A>] - 1 ]</C> will form the additional cell.
For this new cell
requires additional entries are added to the lists <C><A>P</A>.firsts</C> (namely,
<C><A>last</A>+1</C>) and <C><A>P</A>.lengths</C> (namely,
<C><A>P</A>.firsts[<A>m</A>] + <A>P</A>.lengths[<A>m</A>] - <A>last</A> - 1</C>).
</Item>
<Mark>(3)</Mark>
<Item>
The entries of the sublist
<C><A>P</A>.cellno{ [ <A>last</A>+1 .. <A>P</A>.firsts[<A>m</A>] +
P.lengths[<A>m</A>]-1 ] }</C> must be set to the number of the new cell.
</Item>
<Mark>(4)</Mark>
<Item>
The entry <C><A>P</A>.lengths[<A>m</A>]</C> must be reduced to
<C><A>last</A> - <A>P</A>.firsts[<A>m</A>] + 1</C>.
</Item>
</List>
<P/>
Then reuniting the two cells requires only the reversal of steps 2 to 4
above. The list <C><A>P</A>.points</C> need not be rearranged.
</Subsection>
<Subsection Label="Functions for setting up an R-base">
<Heading>Functions for setting up an R-base</Heading>
This subsection explains
some &GAP; functions which are local to the library file
<F>lib/stbcbckt.gi</F> which contains the code for backtracking in permutation
groups. They are mentioned here because you might find them helpful when
you want to implement your own backtracking function based on the
partition concept. An important argument to most of the functions is the
R-base <M>R</M>, which you should regard as a black box. We will tell you how
to set it up, how to maintain it and where to pass it as argument, but it
is not necessary for you to know its internal representation. However, if
you insist to learn the whole story: Here are the record components from
which an R-base is made up:
<P/>
<List>
<Mark><C>domain</C></Mark>
<Item>
the set <M>\Omega</M> on which the group <M>G</M> operates
</Item>
<Mark><C>base</C></Mark>
<Item>
the sequence <M>(a_1, \ldots, a_r)</M> of base points
</Item>
<Mark><C>partition</C></Mark>
<Item>
an ordered partition, initially <M>\Pi_0</M>, this will be refined to
<M>\Pi_1, \ldots, \Pi_r</M> during the backtrack algorithm
</Item>
<Mark><C>where</C></Mark>
<Item>
a list such that <M>a_i</M> lies in cell number <C>where</C><M>[i]</M> of <M>\Pi_i</M>
</Item>
<Mark><C>rfm</C></Mark>
<Item>
a list whose <M>i</M>th entry is a list of refinements which take
<M>\Sigma_i</M> to <M>\Sigma_{{i+1}}</M>; the structure of a refinement is
described below
</Item>
<Mark><C>chain</C></Mark>
<Item>
a (copy of a) stabilizer chain for <M>G</M> (not if <M>G</M> is a symmetric
group)
</Item>
<Mark><C>fix</C></Mark>
<Item>
only if <M>G</M> is a symmetric group: a list whose <M>i</M> entry contains
<C>Fixcells( </C><M>\Pi_i</M><C> )</C>
</Item>
<Mark><C>level</C></Mark>
<Item>
initially equal to <C>chain</C>, this will be changed to chains for the
stabilizers <M>G_{{a_1 \ldots a_i}}</M> for <M>i = 1, \ldots, r</M>
during the
backtrack algorithm; if <M>G</M> is a symmetric group, only the number of
moved points is stored for each stabilizer
</Item>
<Mark><C>lev</C></Mark>
<Item>
a list whose <M>i</M>th entry remembers the <C>level</C> entry for
<M>G_{{a_1 \ldots a_{{i-1}}}}</M>
</Item>
<Mark><C>level2</C>, <C>lev2</C></Mark>
<Item>
a similar construction for a second group (used in intersection
calculations), <K>false</K> otherwise. This second group <M>H</M> activated if
the R-base is constructed as <C>EmptyRBase( </C><M>[ G, H ], \Omega,
\Pi_0</M><C> )</C> (if <M>G = H</M>, &GAP; sets <C>level2 = </C><K>true</K> instead).
</Item>
<Mark><C>nextLevel</C></Mark>
<Item>
this is described below
</Item>
</List>
<P/>
As our guiding example, we present code for the function
<Ref Oper="Centralizer" Label="for a magma and a submagma"/>
which calculates the centralizer of an element <M>g</M> in the group <M>G</M>.
(The real code is more general and has a few more subtleties.)
<P/>
<Listing><![CDATA[
Pi_0 := TrivialPartition( omega );
R := EmptyRBase( G, omega, Pi_0 );
R.nextLevel := function( Pi, rbase )
local fix, p, q, where;
NextRBasePoint( Pi, rbase );
fix := Fixcells( Pi );
for p in fix do
q := p ^ g;
where := IsolatePoint( Pi, q );
if where <> false then
Add( fix, q );
ProcessFixpoint( R, q );
AddRefinement( R, "Centralizer", [ Pi.cellno[ p ], q, where ] );
if Pi.lengths[ where ] = 1 then
p := FixpointCellNo( Pi, where );
ProcessFixpoint( R, p );
AddRefinement( R, "ProcessFixpoint", [ p, where ] );
fi;
fi;
od;
end;
return PartitionBacktrack(
G,
c -> g ^ c = g,
false,
R,
[ Pi_0, g ],
L, R );
]]></Listing>
<P/>
The list numbers below refer to the line numbers of the code above.
<P/>
<List>
<Mark>1.</Mark>
<Item>
<C>omega</C> is the set on which <C>G</C> acts and <C>Pi_0</C>
is the first member of the decreasing sequence of partitions mentioned
in <Ref Sect="Backtrack"/>.
We set <C>Pi_0 = omega</C>, which is constructed as
<C>TrivialPartition( omega )</C>, but we could have started with a finer
partition, e.g., into unions of <C>g</C>-cycles of the same length.
</Item>
<Mark>2.</Mark>
<Item>
This statement sets up the R-base in the variable <C>R</C>.
</Item>
<Mark>3.-21.</Mark>
<Item>
These lines define a function <C>R.nextLevel</C> which is called
whenever an additional member in the sequence
<C>Pi_0 </C><M>\geq \Pi_1 \geq \ldots</M>
of partitions is needed. If <M>\Pi_i</M> does not yet contain enough
base points in one-point cells, &GAP; will call
<C>R.nextLevel( </C><M>\Pi_i,</M><C> R )</C>,
and this function will choose a new base point <M>a_{{i+1}}</M>, refine
<M>\Pi_i</M> to <M>\Pi_{{i+1}}</M>
(thereby <E>changing</E> the first argument) and store
all necessary information in <C>R</C>.
</Item>
<Mark>5.</Mark>
<Item>
This statement selects a new base point <M>a_{{i+1}}</M>,
which is not yet in a one-point cell of <M>\Pi</M> and still moved
by the stabilizer <M>G_{{a_1 \ldots a_i}}</M> of the earlier base points.
If certain points of <C>omega</C> should be preferred as base point
(e.g., because they belong to long cycles of <C>g</C>),
a list of points starting with the most wanted ones, can be given
as an optional third argument to <C>NextRBasePoint</C>
(actually, this is done in the real code for
<Ref Oper="Centralizer" Label="for a magma and a submagma"/>).
</Item>
<Mark>6.</Mark>
<Item>
<C>Fixcells( </C><M>\Pi</M><C> )</C> returns the list of points in
one-point cells of <M>\Pi</M>
(ordered as the cells are ordered in <M>\Pi</M>).
</Item>
<Mark>7.</Mark>
<Item>
For every point <M>p \in fix</M>, if we know the image
<M>p</M><C>^</C><M>g</M> under <M>c \in C_G(e)</M>,
we also know <M>( p</M><C>^</C><M>g )</M><C>^</C><M>c
= ( p</M><C>^</C><M>c )</M><C>^</C><M>g</M>.
We therefore want to isolate these extra points in <M>\Pi</M>.
</Item>
<Mark>9.</Mark>
<Item>
This statement puts point <M>q</M> in a cell of its own,
returning in <C>where</C> the number of the cell of <M>\Pi</M>
from which <M>q</M> was taken.
If <M>q</M> was already the only point in its cell,
<C>where = </C><K>false</K> instead.
</Item>
<Mark>12.</Mark>
<Item>
This command does the necessary bookkeeping for the extra base point
<M>q</M>:
It prescribes <M>q</M> as next base in the stabilizer chain for <M>G</M>
(needed, e.g., in line 5)
and returns <K>false</K> if <M>q</M> was already fixed the
stabilizer of the earlier base points
(and <K>true</K> otherwise; this is not used here).
Another call to <C>ProcessFixpoint</C> like this was implicitly
made by the function <C>NextRBasePoint</C> to register the chosen
base point.
By contrast, the point <M>q</M> was not chosen this way,
so <C>ProcessFixpoint</C> must be called explicitly for <M>q</M>.
</Item>
<Mark>13.</Mark>
<Item>
This statement registers the function which will be used during the
backtrack search to perform the corresponding refinements on the <Q>image
partition</Q> <M>\Sigma_i</M>
(to yield the refined <M>\Sigma_{{i+1}}</M>).
After choosing an image <M>b_{{i+1}}</M> for the base point
<M>a_{{i+1}}</M>, &GAP; will compute
<M>\Sigma_i \wedge (\{ b_{{i+1}} \}, \Omega \setminus \{ b_{{i+1}} \})</M>
and store this partition in <M>I</M><C>.partition</C>,
where <M>I</M> is a black box similar to <M>R</M>,
but corresponding to the current <Q>image partition</Q>
(hence it is an <Q>R-image</Q> in analogy to the R-base).
Then &GAP; will call the function
<C>Refinements.Centralizer( R, I, Pi.cellno[ p ], p, where )</C>,
with the then current values of <M>R</M> and <M>I</M>,
but where <M>\Pi</M><C>.cellno</C><M>[ p ]</M>, <M>p</M>,
<C>where</C> still have the values they have at the time of this
<C>AddRefinement</C> command.
This function call will further refine <M>I</M><C>.partition</C>
to yield <M>\Sigma_{{i+1}}</M> as it is programmed in the function
<C>Refinements.Centralizer</C>, which is described below.
(The global variable <C>Refinements</C> is a record which contains all
refinement functions for all backtracking procedures.)
</Item>
<Mark>14.-19.</Mark>
<Item>
If the cell from which <M>q</M> was taken out had only two points,
we now have an additional one-point cell.
This condition is checked in line 13 and if it is true,
this extra fixpoint <M>p</M> is taken (line 15),
processed like <M>q</M> before (line 16) and is then (line 17)
passed to another refinement function
<C>Refinements.ProcessFixpoint( R, I, p, where )</C>,
which is also described below.
</Item>
<Mark>23.-29.</Mark>
<Item>
This command starts the backtrack search. Its result will be the
centralizer as a subgroup of <M>G</M>. Its arguments are
</Item>
<Mark>24.</Mark>
<Item>
the group we want to run through,
</Item>
<Mark>25.</Mark>
<Item>
the property we want to test, as a &GAP; function,
</Item>
<Mark>26.</Mark>
<Item>
<K>false</K> if we are looking for a subgroup, <K>true</K> in the case
of a representative search (when the result would be one
representative),
</Item>
<Mark>27.</Mark>
<Item>
the R-base,
</Item>
<Mark>28.</Mark>
<Item>
a list of data, to be stored in <M>I</M><C>.data</C>, which has
in position 1 the first member <M>\Sigma_0</M> of the decreasing
sequence of <Q>image partitions</Q> mentioned in
<Ref Sect="Backtrack"/>.
In the centralizer example, position 2 contains the element that is to be centralized. In the case of a representative
search, i.e., a conjugacy test <M>g</M><C>^</C><M>c</M><C> ?= </C><M>h</M>, we
would have <M>h</M> instead of <M>g</M> here, and possibly a <M>\Sigma_0</M>
different from <M>\Pi_0</M> (e.g., a partition into unions of <M>h</M>-cycles
of same length).
</Item>
<Mark>29.</Mark>
<Item>
two subgroups <M>L \leq C_G(g)</M> and <M>R \leq C_G(h)</M> known in
advance (we have <M>L = R</M> in the centralizer case).
</Item>
</List>
</Subsection>
<Subsection Label="Refinement functions for the backtrack search">
<Heading>Refinement functions for the backtrack search</Heading>
The last
subsection showed how the refinement process leading from <M>\Pi_i</M> to
<M>\Pi_{{i+1}}</M> is coded in the function <M>R</M><C>.nextLevel</C>, this has to be
executed once the base point <M>a_{{i+1}}</M>. The analogous refinement step
from <M>\Sigma_i</M> to <M>\Sigma_{{i+1}}</M> must be performed for each choice of an
image <M>b_{{i+1}}</M> for <M>a_{{i+1}}</M>, and it will depend on the corresponding
value of <M>\Sigma_i \wedge (\{b_{{i+1}}\}, \Omega \setminus \{b_{{i+1}}\})</M>.
But before
we can continue our centralizer example, we must, for the interested
reader, document the record components of the other black box <M>I</M>, as we
did above for the R-base black box <M>R</M>. Most of the components change as
&GAP; walks up and down the levels of the search tree.
<List>
<Mark><C>data</C></Mark>
<Item>
this will be mentioned below
</Item>
<Mark><C>depth</C></Mark>
<Item>
the level <M>i</M> in the search tree of the current node <M>\Sigma_i</M>
</Item>
<Mark><C>bimg</C></Mark>
<Item>
a list of images of the points in <M>R</M><C>.base</C>
</Item>
<Mark><C>partition</C></Mark>
<Item>
the partition <M>\Sigma_i</M> of the current node
</Item>
<Mark><C>level</C></Mark>
<Item>
the stabilizer chain <M>R</M><C>.lev</C><M>[i]</M> at the current level
</Item>
<Mark><C>perm</C></Mark>
<Item>
a permutation mapping <C>Fixcells</C><M>( \Pi_i )</M> to
<C>Fixcells</C><M>( \Sigma_i )</M>;
this implies mapping <M>(a_1, \ldots, a_i)</M> to
<M>(b_1, \ldots, b_i)</M>
</Item>
<Mark><C>level2</C>, <C>perm2</C></Mark>
<Item>
a similar construction for the second stabilizer chain, <K>false</K>
otherwise (and <K>true</K> if <M>R</M><C>.level2 = </C><K>true</K>)
</Item>
</List>
<P/>
As declared in the above code for <Ref Oper="Centralizer" Label="for a magma and a submagma"/>,
the refinement is performed by the function
<C>Refinement.Centralizer</C><M>( R, I, \Pi</M><C>.cellno</C><M>[p], p, where )</M>.
The functions in the record
<C>Refinement</C> always take two additional arguments before the ones
specified in the <C>AddRefinement</C> call (in line 13 above), namely the
R-base <M>R</M> and the current value <M>I</M> of the <Q>R-image</Q>.
In our example, <M>p</M> is a fixpoint of
<M>\Pi = \Pi_i \wedge (\{ a_{{i+1}} \}, \Omega \setminus \{ a_{{i+1}} \})</M>
such that <M>where = \Pi</M><C>.cellno</C><M>[ p^g ]</M>.
The <C>Refinement</C> functions must return <K>false</K> if the refinement is
unsuccessful (e.g., because it leads to <M>\Sigma_{{i+1}}</M> having different
cell sizes from <M>\Pi_{{i+1}}</M>) and <K>true</K> otherwise.
Our particular function looks like this.
<P/>
<Listing><![CDATA[
Refinements.Centralizer := function( R, I, cellno, p, where )
local Sigma, q;
Sigma := I.partition;
q := FixpointCellNo( Sigma, cellno ) ^ I.data[ 2 ];
return IsolatePoint( Sigma, q ) = where and ProcessFixpoint( I, p, q );
end;
]]></Listing>
<P/>
The list numbers below refer to the line numbers of the code immediately
above.
<P/>
<List>
<Mark>3.</Mark>
<Item>
The current value of
<M>\Sigma_i \wedge (\{ b_{{i+1}} \}, \Omega \setminus \{ b_{{i+1}} \})</M>
is always found in <M>I</M><C>.partition</C>.
</Item>
<Mark>4.</Mark>
<Item>
The image of the only point in cell number
<M>cellno = \Pi_i</M><C>.cellno</C><M>[ p ]</M> in <M>\Sigma</M>
under <M>g = I</M><C>.data</C><M>[ 2 ]</M> is calculated.
</Item>
<Mark>5.</Mark>
<Item>
The function returns <K>true</K> only if the image <M>q</M> has the
same cell number in <M>\Sigma</M> as <M>p</M> had in <M>\Pi</M>
(i.e., <M>where</M>) and if <M>q</M> can be prescribed as an image
for <M>p</M> under the coset of the stabilizer
<M>G_{{a_1 \ldots a_{{i+1}}}}.c</M> where <M>c \in G</M> is an
(already constructed) element mapping the earlier base points
<M>a_1, \ldots, a_{{i+1}}</M> to the already chosen images
<M>b_1, \ldots, b_{{i+1}}</M>.
This latter condition is tested by
<C>ProcessFixpoint</C><M>( I, p, q )</M> which, if successful,
also does the necessary bookkeeping in <M>I</M>.
In analogy to the remark about line 12 in the program above,
the chosen image <M>b_{{i+1}}</M> for the base point <M>a_{{i+1}}</M>
has already been processed implicitly by the function
<C>PartitionBacktrack</C>,
and this processing includes the construction of an element
<M>c \in G</M> which maps <C>Fixcells</C><M>( \Pi_i )</M> to
<C>Fixcells</C><M>( \Sigma_i )</M> and <M>a_{{i+1}}</M> to
<M>b_{{i+1}}</M>.
By contrast, the extra fixpoints <M>p</M> and <M>q</M> in
<M>\Pi_{{i+1}}</M> and <M>\Sigma_{{i+1}}</M> were not chosen automatically,
so they require an explicit call of <C>ProcessFixpoint</C>,
which replaces the element <M>c</M> by some <M>c'.c
(with <M>c' \in G_{{a_1 \ldots a_{{i+1}}}}) which in addition maps
<M>p</M> to <M>q</M>, or returns <K>false</K> if this is impossible.
</Item>
</List>
<P/>
You should now be able to guess what
<C>Refinements.ProcessFixpoint</C><M>( R, I, p, where )</M> does:
it simply returns
<C>ProcessFixpoint</C><M>( I, p, </M><C>FixpointCellNo</C><M>(
I</M><C>.partition</C><M>, where ) )</M>.
<P/>
<E>Summary.</E>
<P/>
When you write your own backtrack functions using
the partition technique, you have to supply an R-base, including a
component <C>nextLevel</C>, and the functions in the <C>Refinements</C> record
which you need. Then you can start the backtrack by passing the R-base
and the additional data (for the <C>data</C> component of the <Q>R-image</Q>) to
<C>PartitionBacktrack</C>.
</Subsection>
<Subsection Label="Functions for meeting ordered partitions">
<Heading>Functions for meeting ordered partitions</Heading>
A kind of
refinement that occurs in particular in the normalizer calculation
involves computing the meet of <M>\Pi</M> (cf. lines 6ff. above) with an
arbitrary other partition <M>\Lambda</M>, not just with one point. To do this
efficiently, &GAP; uses the following two functions.
<P/>
<C>StratMeetPartition( </C><M>R</M>, <M>\Pi</M>, <M>\Lambda</M> <C>[</C>, <M>g</M> <C>] )</C>
<P/>
<C>MeetPartitionStrat( </C><M>R</M>, <M>I</M><C>{, </C><M>\Lambda'}[, {g'</M><C>}]</C>, <M>strat</M> <C>)</C>
<P/>
<Index>meet strategy</Index>
Such a <C>StratMeetPartition</C> command would typically appear in the
function call <M>R</M><C>.nextLevel</C><M>( \Pi, R )</M> (during the refinement of
<M>\Pi_i</M> to <M>\Pi_{{i+1}}</M>).
This command replaces <M>\Pi</M> by <M>\Pi \wedge \Lambda</M>
(thereby <E>changing</E> the second argument) and returns a <Q>meet
strategy</Q> <M>strat</M>. This is (for us) a black box which serves two
purposes: First, it allows &GAP; to calculate faster the corresponding
meet <M>\Sigma \wedge \Lambda', which must then appear in a Refinements
function (during the refinement of <M>\Sigma_i</M> to <M>\Sigma_{{i+1}}</M>). It is
faster to compute <M>\Sigma \wedge \Lambda' with the meet strategy of
<M>\Pi \wedge \Lambda</M> because if the refinement of <M>\Sigma</M> is successful
at all, the intersection of a cell from the left hand side of the
<M>\wedge</M> sign with a cell from the right hand side must have the same
size in both cases (and <M>strat</M> records these sizes, so that only
non-empty intersections must be calculated for <M>\Sigma \wedge \Lambda').
Second, if there is a discrepancy between the behaviour prescribed by
<M>strat</M> and the behaviour observed when refining <M>\Sigma</M>, the refinement
can immediately be abandoned.
<P/>
On the other hand, if you only want to meet a partition <M>\Pi</M> with
<M>\Lambda</M> for a one-time use, without recording a strategy, you can
simply type <C>StratMeetPartition</C><M>( \Pi, \Lambda )</M> as in the following
example, which also demonstrates some other partition-related commands.
<P/> <!-- % <Example><![CDATA[ --> <!-- % gap> P := Partition( [[1,2],[3,4,5],[6]] );; Cells( P ); --> <!-- % [ [ 1, 2 ], [ 3, 4, 5 ], [ 6 ] ] --> <!-- % gap> Q := OnPartitions( P, (1,3,6) );; Cells( Q ); --> <!-- % [ [ 3, 2 ], [ 6, 4, 5 ], [ 1 ] ] --> <!-- % gap> StratMeetPartition( P, Q ); --> <!-- % [ ] # the ``meet strategy'' was not recorded, ignore this result --> <!-- % gap> Cells( P ); --> <!-- % [ [ 1 ], [ 5, 4 ], [ 6 ], [ 2 ], [ 3 ] ] --> <!-- % ]]></Example> --> <!-- --> <!-- % The preceding (original) example doesn't work because there is no --> <!-- % function 'OnPartitions'. The following example works, but I don't --> <!-- % know if it makes sense to have it here at all. (Note, in particular, --> <!-- % that the function 'Partition' is undocumented.) VF 14.10.02 -->
<Example><![CDATA[
gap> P := Partition( [[1,2],[3,4,5],[6]] );; Cells( P );
[ [ 1, 2 ], [ 3, 4, 5 ], [ 6 ] ]
gap> Q := Partition( OnTuplesTuples( last, (1,3,6) ) );; Cells( Q );
[ [ 3, 2 ], [ 6, 4, 5 ], [ 1 ] ]
gap> StratMeetPartition( P, Q );
[ ]
gap> # The ``meet strategy'' was not recorded, ignore this result.
gap> Cells( P );
[ [ 1 ], [ 5, 4 ], [ 6 ], [ 2 ], [ 3 ] ]
]]></Example>
<P/>
You can even say <C>StratMeetPartition</C><M>( \Pi, \Delta )</M> where <M>\Delta</M>
is simply a subset of <M>\Omega</M>, it will then be interpreted as the
partition <M>(\Delta, \Omega \setminus \Delta)</M>.
<P/>
&GAP; makes use of the advantages of a <Q>meet strategy</Q> if the
refinement function in <C>Refinements</C> contains a <C>MeetPartitionStrat</C>
command where <M>strat</M> is the <Q>meet strategy</Q> calculated by
<C>StratMeetPartition</C> before. Such a command replaces <M>I</M><C>.partition</C> by
its meet with <M>\Lambda', again changing the argument I. The necessary
reversal of these changes when backtracking from a node (and prescribing
the next possible image for a base point) is automatically done by the
function <C>PartitionBacktrack</C>.
<P/>
In all cases, an additional argument <M>g</M> means that the meet is to be
taken not with <M>\Lambda</M>, but instead with <M>\Lambda.{{g^{{-1}}}}</M>,
where
operation on ordered partitions is meant cellwise (and setwise on each
cell). (Analogously for the primed arguments.)
<P/>
<Example><![CDATA[
gap> P := Partition( [[1,2],[3,4,5],[6]] );;
gap> StratMeetPartition( P, P, (1,6,3) );; Cells( P );
[ [ 1 ], [ 5, 4 ], [ 6 ], [ 2 ], [ 3 ] ]
]]></Example>
Note that <M>P.(1,3,6) = Q</M>.
</Subsection>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Stabilizer Chains for Automorphisms Acting on Enumerators">
<Heading>Stabilizer Chains for Automorphisms Acting on Enumerators</Heading>
This section describes a way of representing the automorphism group of a
group as permutation group, following <Cite Key="Sims97"/>. The code however is
not yet included in the &GAP; library.
<P/>
In this section we present an example in which objects we already know
(namely, automorphisms of solvable groups) are equipped with the
permutation-like operations <C>^</C> and <C>/</C> for action on positive integers.
To achieve this, we must define a new type of objects which behave like
permutations but are represented as automorphisms acting on an
enumerator. Our goal is to generalize the Schreier-Sims algorithm for
construction of a stabilizer chain to groups of such new automorphisms.
<Subsection Label="An operation domain for automorphisms">
<Heading>An operation domain for automorphisms</Heading>
The idea we describe here is due to C. Sims.
We consider a group <M>A</M> of automorphisms of a
group <M>G</M>, given by generators, and we would like to know its order. Of
course we could follow the strategy of the Schreier-Sims algorithm
(described in <Ref Sect="Stabilizer Chains"/>) for <M>A</M>
acting on <M>G</M>. This would involve a call of
<C>StabChainStrong( EmptyStabChain( [], One( <A>A</A> ) ), GroupGenerators( <A>A</A> ) )</C> where
<C>StabChainStrong</C> is a function as the one described in the pseudo-code
below:
<P/>
<Listing><![CDATA[
StabChainStrong := function( S, newgens )
Extend the Schreier tree of S with newgens.
for sch in Schreier generators do
if not sch in S.stabilizer then
StabChainStrong( S.stabilizer, [ sch ] );
fi;
od;
end;
]]></Listing>
<P/>
The membership test <M>sch \notin S</M><C>.stabilizer</C> can be performed because
the stabilizer chain of <M>S</M><C>.stabilizer</C> is already correct at that
moment. We even know a base in advance, namely any generating set for
<M>G</M>.
Fix such a generating set <M>(g_1, \ldots, g_d)</M> and observe that this
base is generally very short compared to the degree <M>|G|</M> of the
operation. The problem with the Schreier-Sims algorithm, however, is then
that the length of the first basic orbit <M>g_1.A</M> would already have the
magnitude of <M>|G|</M>, and the basic orbits at deeper levels would not be
much shorter. For the advantage of a short base we pay the high price of
long basic orbits, since the product of the (few) basic orbit lengths
must equal <M>|A|</M>. Such long orbits make the Schreier-Sims algorithm
infeasible, so we have to look for a longer base with shorter basic
orbits.
<P/>
Assume that <M>G</M> is solvable and choose a characteristic series with
elementary abelian factors. For the sake of simplicity we assume that <M>N
< G</M> is an elementary abelian characteristic subgroup with elementary
abelian factor group <M>G/N</M>. Since <M>N</M> is characteristic, <M>A</M> also acts as
a group of automorphisms on the factor group <M>G/N</M>, but of course not
necessarily faithfully. To retain a faithful action, we let <M>A</M> act on
the disjoint union <M>G/N</M> with <M>G</M>, and choose as base
<M>(g_1 N, \ldots, g_d N, g_1, \ldots, g_d)</M>.
Now the first <M>d</M> basic orbits lie
inside <M>G/N</M> and can have length at most <M>[G:N]</M>. Since the base
points <M>g_1 N, \ldots, g_d N</M> form a generating set for <M>G/N</M>, their
iterated stabilizer <M>A^{(d+1)}</M> acts trivially on the factor group <M>G/N</M>,
i.e., it leaves the cosets <M>g_i N</M> invariant. Accordingly, the next <M>d</M>
basic orbits lie inside <M>g_i N</M> (for <M>i = 1, \ldots, d</M>) and can have length
at most <M>|N|</M>.
<P/>
Generalizing this method to a characteristic series
<M>G = N_0 > N_1 > \ldots > N_l = \{ 1 \}</M> of length
<M>l > 2</M>, we can always find a base of length <M>l.d</M>
such that each basic orbit is contained in a coset of a characteristic
factor, i.e. in a set of the form <M>g_i N_{{j-1}} / N_j</M>
(where <M>g_i</M> is one of
the generators of <M>G</M> and <M>1 \leq j \leq l</M>).
In particular, the length of
the basic orbits is bounded by the size of the corresponding
characteristic factors. To implement a Schreier-Sims algorithm for such a
base, we must be able to let automorphisms act on cosets of
characteristic factors <M>g_i N_{{j-1}} / N_j</M>,
for varying <M>i</M> and <M>j</M>. We
would like to translate each such action into an action on
<M>\{ 1, \ldots, [ N_{{j-1}}:N_j] \}</M>,
because then we need not enumerate the operation domain,
which is the disjoint union of
<M>G / N_1</M>, <M>G / N_2 \ldots G / N_l</M>,
as a whole. Enumerating it as a whole would result in basic orbits like
<C>orbit</C><M> \subseteq \{ 1001, \ldots, 1100 \}</M>
with a <C>transversal</C> list whose
first 1000 entries would be unbound, but still require 4 bytes of memory
each (see <Ref Sect="Stabilizer Chain Records"/>).
<P/>
Identifying each coset <M>g_i N_{{j-1}} / N_j</M> into
<M>\{ 1, \ldots, [N_{{j-1}}:N_j] \}</M> of course means
that we have to change the action of
the automorphisms on every level of the stabilizer chain. Such
flexibility is not possible with permutations because their effect on
positive integers is <Q>hardwired</Q> into them, but we can install new
operations for automorphisms.
</Subsection>
<Subsection Label="Enumerators for cosets of characteristic factors">
<Heading>Enumerators for cosets of characteristic factors</Heading>
So far we
have not used the fact that the characteristic factors are elementary
abelian, but we will do so from here on.
Our first task is to implement an enumerator
(see <Ref Attr="AsList"/> and
<Ref Sect="Enumerators"/>)
for a coset of a characteristic factor in a solvable group <M>G</M>.
We assume that such a coset <M>g N/M</M> is given by
<P/>
<List>
<Mark>(1)</Mark>
<Item>
a pcgs for the group <M>G</M> (see <Ref Attr="Pcgs"/>),
let <M>n = </M><C>Length( </C><M>pcgs</M><C> )</C>;
</Item>
<Mark>(2)</Mark>
<Item>
a range <M>range = [ start .. stop ]</M> indicating that
<M>N = \langle pcgs\{ [ start .. n ] \} \rangle</M> and
<M>M = \langle pcgs\{ [ stop + 1 .. n ] \} \rangle</M>,
i.e., the cosets of <M>pcgs\{ range \}</M> form a base
for the vector space <M>N/M</M>;
</Item>
<Mark>(3)</Mark>
<Item>
the representative <M>g</M>.
</Item>
</List>
<P/>
We first define a new representation for such enumerators and then
construct them by simply putting these three pieces of data into a record
object. The enumerator should behave as a list of group elements
(representing cosets modulo <M>M</M>), consequently, its family will be the
family of the <M>pcgs</M> itself.
<P/>
<Log><![CDATA[
DeclareRepresentation( "IsCosetSolvableFactorEnumeratorRep", IsEnumerator,
[ "pcgs", "range", "representative" ] );
EnumeratorCosetSolvableFactor := function( pcgs, range, g )
return Objectify( NewType( FamilyObj( pcgs ),
IsCosetSolvableFactorEnumeratorRep ),
rec( pcgs := pcgs,
range := range,
representative := g ) );
end;
]]></Log>
<P/>
The definition of the operations <Ref Attr="Length"/>,
<Ref Oper="\[\]"/> and <Ref Oper="Position"/>
is now
straightforward. The code has sometimes been abbreviated and is meant
<Q>cum grano salis</Q>, e.g., the declaration of the local variables has
been left out.
<P/>
<Log><![CDATA[
InstallMethod( Length, [ IsCosetSolvableFactorEnumeratorRep ],
enum -> Product( RelativeOrdersPcgs( enum!.pcgs ){ enum!.range } ) );
InstallMethod( \[\], [ IsCosetSolvableFactorEnumeratorRep,
IsPosRat and IsInt ],
function( enum, pos )
elm := ();
pos := pos - 1;
for i in Reversed( enum!.range ) do
p := RelativeOrderOfPcElement( enum!.pcgs, i );
elm := enum!.pcgs[ i ] ^ ( pos mod p ) * elm;
pos := QuoInt( pos, p );
od;
return enum!.representative * elm;
end );
InstallMethod( Position, [ IsCosetSolvableFactorEnumeratorRep,
IsObject, IsZeroCyc ],
function( enum, elm, zero )
exp := ExponentsOfPcElement( enum!.pcgs,
LeftQuotient( enum!.representative, elm ) );
pos := 0;
for i in enum!.range do
pos := pos * RelativeOrderOfPcElement( pcgs, i ) + exp[ i ];
od;
return pos + 1;
end );
]]></Log>
</Subsection>
<Subsection Label="Making automorphisms act on such enumerators">
<Heading>Making automorphisms act on such enumerators</Heading>
Our next task is to make automorphisms of the solvable group
<M>pcgs</M><C>!.group</C> act on
<M>[ 1 .. </M><C>Length</C><M>( enum ) ]</M> for such an enumerator
<M>enum</M>. We achieve this
by introducing a new representation of automorphisms on enumerators and
by putting the enumerator together with the automorphism into an object
which behaves like a permutation. Turning an ordinary automorphism into
such a special automorphism requires then the construction of a new
object which has the new type. We provide an operation
<C>PermOnEnumerator( <A>model</A>, <A>aut</A> )</C> which constructs such a new
object having the same type as <A>model</A>,
but representing the automorphism <A>aut</A>. So <A>aut</A> can be
either an ordinary automorphism or one which already has an enumerator in
its type, but perhaps different from the one we want (i.e. from the one
in <A>model</A>).
<P/>
<Log><![CDATA[
DeclareCategory( "IsPermOnEnumerator",
IsMultiplicativeElementWithInverse and IsPerm );
DeclareRepresentation( "IsPermOnEnumeratorDefaultRep",
IsPermOnEnumerator and IsAttributeStoringRep,
[ "perm" ] );
InstallMethod( PermOnEnumerator,
[ IsEnumerator, IsObject ],
function( enum, a )
SetFilterObj( a, IsMultiplicativeElementWithInverse );
a := Objectify( NewKind( PermutationsOnEnumeratorsFamily,
IsPermOnEnumeratorDefaultRep ),
rec( perm := a ) );
SetEnumerator( a, enum );
return a;
end );
InstallMethod( PermOnEnumerator,
[ IsEnumerator, IsPermOnEnumeratorDefaultRep ],
function( enum, a )
a := Objectify( TypeObj( a ), rec( perm := a!.perm ) );
SetEnumerator( a, enum );
return a;
end );
]]></Log>
<P/>
Next we have to install new methods for the operations which calculate
the product of two automorphisms, because this product must again have
the right type. We also have to write a function which uses the
enumerators to apply such an automorphism to positive integers.
<P/>
<Log><![CDATA[
InstallMethod( \*, IsIdenticalObj,
[ IsPermOnEnumeratorDefaultRep, IsPermOnEnumeratorDefaultRep ],
function( a, b )
perm := a!.perm * b!.perm;
SetIsBijective( perm, true );
return PermOnEnumerator( Enumerator( a ), perm );
end );
InstallMethod( \^,
[ IsPosRat and IsInt, IsPermOnEnumeratorDefaultRep ],
function( p, a )
return PositionCanonical( Enumerator( a ),
Enumerator( a )[ p ] ^ a!.perm );
end );
]]></Log>
<P/>
How the corresponding methods for <C><A>p</A> / <A>aut</A></C>
and <C><A>aut</A> ^ <A>n</A></C> look like is obvious.
<P/>
Now we can formulate the recursive procedure <C>StabChainStrong</C> which
extends the stabilizer chain by adding in new generators <M>newgens</M>. We
content ourselves again with pseudo-code, emphasizing only the lines
which set the <C>EnumeratorDomainPermutation</C>. We assume that initially <M>S</M>
is a stabilizer chain for the trivial subgroup with a level for each pair
<M>(range,g)</M> characterizing an enumerator (as described above). We also
assume that the <C>identity</C> element at each level already has the type
corresponding to that level.
<P/>
<Listing><![CDATA[
StabChainStrong := function( S, newgens )
for i in [ 1 .. Length( newgens ) ] do
newgens[ i ] := AutomorphismOnEnumerator( S.identity, newgens[ i ] );
od;
Extend the Schreier tree of S with newgens.
for sch in Schreier generators do
if not sch in S.stabilizer then
StabChainStrong( S.stabilizer, [ sch ] );
fi;
od;
end;
]]></Listing>
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