search.xml orb package documentation Juergen Mueller and Max Neunhoeffer and Felix Noeske
Copyright (C) 2005-2008 by the authors
This chapter explains functionality for searching in groups and orbits.
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<Chapter Label="search">
<Heading>Searching in groups and orbits</Heading>
<Section>
<Heading>Searching using orbit enumeration</Heading>
As described in Subsection <Ref Subsect="orboptions"/> the standard
orbit enumeration routines can perform search operations during orbit enumeration. If one is looking for a shortest word in the generators
of a group to express a group element with a certain property, then
this natural breadth-first search can be used, for example by letting
the group act on its own elements, either by multiplication or by
conjugation.
<P/>
All technical instructions to do this are already given in Subsection
<Ref Subsect="orboptions"/>, so we are content to provide an example
here. Assume you want to find the shortest way to express some
<M>7</M>-cycle in the symmetric group <M>S_{{10}}</M> as a product of
generators <M>a := </M><C>(1,2,3,4,5,6,7,8,9,10)</C> and
<M>b := </M><C>(1,2)</C>.
Then you could do this in the following way:
<Example><![CDATA[gap> gens := [(1,2,3,4,5,6,7,8,9,10),(1,2)];
[ (1,2,3,4,5,6,7,8,9,10), (1,2) ]
gap> o := Orb(gens,(),OnRight,rec( report := 2000,
> lookingfor :=
> function(o,x) return NrMovedPoints(x) = 7 and Order(x)=7; end,
> schreier := true ));
<open orbit, 1 points with Schreier tree looking for sth.>
gap> Enumerate(o);
<open orbit, 614 points with Schreier tree looking for sth.>
gap> w := TraceSchreierTreeForward(o,PositionOfFound(o));
[ 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2 ]
gap> ActWithWord(o!.gens,w,o!.op,o[1]);
(1,10,9,8,7,6,5)
gap> o[PositionOfFound(o)] = ActWithWord(o!.gens,w,o!.op,o[1]);
true
gap> EvaluateWord(o!.gens,w);
(1,10,9,8,7,6,5)]]></Example>
The result shows that <M>a^6 \cdot (a\cdot b)^3</M> is a <M>7</M>-cycle
and that there is no word in <M>a</M> and <M>b</M> with fewer than <M>12</M>
letters expressing a <M>7</M>-cycle.
<P/>
Note that we can go on with the above orbit enumeration to find further
words to express <M>7</M>-cycles.
</Section>
<Section>
<Heading>Random searches in groups</Heading>
Another possibility to look for elements in a group satisfying certain
properties is to look at random elements, usually obtained by doing
product replacement (see Section <Ref Sect="prodrepl"/>). Although this
can lead to very long expressions as words in the generators, one can
cope with this problem by using the <C>maxdepth</C> option of the
product replacer objects, which just reinitialises the product replacement
table after a certain number of product replacements has been performed.
To organise all this conveniently, there is the concept of
<Q>random searcher objects</Q> described here.
<ManSection>
<Oper Name="RandomSearcher" Arg="gens, testfunc [,opt]"/>
<Returns> a random searcher object </Returns>
<Description>
<A>gens</A> must be a list of group generators, <A>testfunc</A> a function
taking as argument one group element and returning <K>true</K> or <K>false</K>.
<A>opt</A> is an optional options record. For possible options see below.
<P/>
At first, the random searcher object is just initialised but no random
searching is performed. The actual search is triggered by
the <Ref Oper="Search"/> operation (see below). The idea of
random searcher objects is that a product replacer object is
created and during a search random elements are produced using
this product replacer object, and for each group element produced
the function <A>testfunc</A> is called. If this function returns
<K>true</K>, the search stops and the group element found is returned.
<P/>
The following options can be bound in the options record <A>opt</A>:
<List>
<Mark><C>exceptions</C></Mark>
<Item>This component can be a list to initialise the exception list
in the random searcher object. Group elements in this list are not
considered as successful searches. This list is also
used to continue search operations to found more suitable
group elements as every group element considered <Q>found</Q> is
added to that list before returning it. Thus every element will
only be found once.
</Item>
<Mark><C>maxdepth</C></Mark>
<Item>Sets the <C>maxdepth</C> option of the created product replacer object.
This limits the length of the expression as product of the generators
of the found group elements.
</Item>
<Mark><C>addslots</C></Mark>
<Item>Sets the <C>addslots</C> option of the created product replacer object.
</Item>
<Mark><C>scramble</C></Mark>
<Item>If this component is bound at all, then the created product replacer
object is created with options <C>scramble=100</C> and
<C>scramblefactor=10</C> (the default values), otherwise the options
<C>scramble=0</C> and <C>scramblefactor=0</C> are used, leading to
no scrambling at all. See <Ref Oper="ProductReplacer"/> for
details on the product replacement algorithm.
</Item>
</List>
Note that of course the generators in <A>gens</A> may have memory.
However, the function <A>testfunc</A> is called with the group element
with memory stripped off.
</Description>
</ManSection>
<ManSection>
<Oper Name="Search" Arg="rs"/>
<Returns> a group element </Returns>
<Description>
Runs the search with the random searcher object <A>rs</A> and returns
with the first group element found.
</Description>
</ManSection>
</Section>
<Section>
<Heading>The dihedral trick and applications</Heading>
With the <Q>dihedral</Q> trick we mean the following: Two involutions
<M>a</M> and <M>b</M> in a group always generate a dihedral group.
Thus, to find a pseudo-random element in the centraliser of
an involution <M>a</M>, we can proceed as follows: Create a
pseudo-random element <M>c</M>, then <M>b := a^c</M> is another
involution. If then <M>ab</M> has order <M>2o</M>, we can use
<M>(ab)^o</M>. Otherwise, if the order of <M>ab</M> is <M>2o-1</M>,
then <M>(ab)^o \cdot c^{{-1}}</M> centralises <M>a</M>.
<P/>
This trick allows to efficiently produce elements in the centraliser
of an involution and thus, with high probability, generators for the
full centraliser.
<P/>
There are the following functions:
<ManSection>
<Func Name="FindInvolution" Arg="pr"/>
<Returns> an involution </Returns>
<Description>
<A>pr</A> must be a product replacer object (see Section
<Ref Sect="prodrepl"/>). Searches an involution by finding a random element of even order and powering up. Returns the involution.
</Description>
</ManSection>
<ManSection>
<Func Name="FindCentralisingElementOfInvolution" Arg="pr, a"/>
<Returns> a group element </Returns>
<Description>
<A>pr</A> must be a product replacer object (see Section
<Ref Sect="prodrepl"/>). Produces one random element and builds
an element the centralises the involution <A>a</A> using the
dihedral trick described above.
</Description>
</ManSection>
<ManSection>
<Func Name="FindInvolutionCentralizer" Arg="pr, a, nr"/>
<Returns> a list of <A>nr</A> group elements </Returns>
<Description>
<A>pr</A> must be a product replacer object (see Section
<Ref Sect="prodrepl"/>) and <A>a</A> and involution. This function
uses <Ref Func="FindCentralisingElementOfInvolution"/> <A>nr</A> times
to produce an element centralising the involution <A>a</A> and returns
the list of results.
</Description>
</ManSection>
</Section>
<Section>
<Heading>Orbit statistics on vector spaces</Heading>
The following two functions are tools to get a rough and quick estimate
about the average orbit length of a group acting on a vector space.
<ManSection>
<Func Name="OrbitStatisticOnVectorSpace" Arg="gens, size, ti"/>
<Returns> nothing </Returns>
<Description>
<A>gens</A> must be a list of matrix group generators and <A>size</A>
an integer giving an upper bound for the lengths of orbits (for example
given by the order of the group generated by <A>gens</A>). <A>ti</A>
is an integer specifying the number of seconds to run. This function
enumerates orbits of random vectors in the natural space the group is
acting on (with an upper limit of length given by <A>size</A>). The
average length and some other information is printed on the screen.
</Description>
</ManSection>
<ManSection>
<Func Name="OrbitStatisticOnVectorSpaceLines" Arg="gens, size, ti"/>
<Returns> nothing </Returns>
<Description>
<A>gens</A> must be a list of matrix group generators and <A>size</A> an
integer giving an upper bound for the lengths of orbits (for example the
order of the group generated by <A>gens</A>). <A>ti</A> is an integer
specifying the number of seconds to run. This function enumerates orbits
of random one-dimensional subspaces in the natural space the group is
acting on (with an upper limit of length given by <A>size</A>). The
average length and some other information is printed on the screen.
</Description>
</ManSection>
</Section>
<Section>
<Heading>Finding generating sets of subgroups</Heading>
The following function can be used to find generators of a subgroup
of a group <M>G</M>, expressed as a straight line program in the generators
of <M>G</M>.
<ManSection>
<Meth Name="FindShortGeneratorsOfSubgroup" Arg="G, U [, membopt]"/>
<Returns> a record described below </Returns>
<Description>
The arguments <A>U</A> and <A>G</A> must be &GAP; group objects with
<A>U</A> being a subgroup of <A>G</A>. The argument
<A>membopt</A> can be a function taking two arguments, namely
a group element and a group, that checks, whether the group element
lies in the group or not, returning <K>true</K> or <K>false</K>
accordingly. You can usually just use the function <C>&bslash;in</C>
as third argument. Note that this function will only ever be called
with <A>U</A> as its second argument so you can in fact provide
a function which ignores its second argument and has <A>U</A> somehow
built in it.
<P/>
Optionally, the third argument <A>membopt</A> can also be an options
record. The component <C>membershiptest</C> has the same meaning like
the third argument <A>membopt</A> above. The component
<C>sizetester</C> can be bound to a function which estimates the size
of a group generated by some elements in <A>U</A>. This estimate
function can for example be a function which runs a random Schreier-Sims
algorithm. In particular it may underestimate the size with a certain
probability. The method <Ref Meth="FindShortGeneratorsOfSubgroup"/>
will keep looking for elements to generate <A>U</A> until the
<C>sizetester</C> returns the same number as for <A>U</A> itself.
Note that to avoid the possibility that the <C>sizetester</C>
underestimates the size of <A>U</A> in the first place you can bind the
component <C>sizeU</C> in the options record to the correct size of
<A>U</A> or make sure that the group object <A>U</A> does know its size
before the call to <Ref Meth="FindShortGeneratorsOfSubgroup"/>.
<P/>
This function does a breadth-first search to find elements in <A>U</A>
using the generators of <A>G</A>. It then uses calculates the size of
the group generated and proceeds in this way, until a generating
system for <A>U</A> is found in terms of the generators of <A>G</A>. Those
generators are guaranteed to be shortest words in the generators
of <A>G</A> lying in <A>U</A>.
<P/>
The function returns a record with two components bound: <C>gens</C>
is a list of generators for <A>U</A> and <C>slp</C> is a &GAP; straight
line program expressing exactly those generators in the generators
of <A>G</A>.
<P/>
Note that this function only performs satisfactorily when the index
of <A>U</A> in <A>G</A> is not to huge. It also helps if the groups
come in a representation in which &GAP; can compute efficiently for
example as permutation groups.
</Description>
</ManSection>
</Section>
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