Quelle permutat.xml Sprache: XML

<Chapter Label="Permutations">
<Heading>Permutations</Heading>

&GAP; offers a data type <E>permutation</E> to describe the elements
of permutation groups.
<P/>
The points on which permutations in &GAP; act are the positive
integers up to a certain architecture dependent limit,
and the image of a point <M>i</M> under a permutation <M>p</M> is written
<M>i^p</M>, which is expressed as <M>i</M><C>^</C><M>p</M> in &GAP;.
(This action is also implemented by the function <Ref Func="OnPoints"/>.)
If <M>i</M><C>^</C><M>p</M> is different from <M>i</M>,
we say that <M>i</M> is <E>moved</E> by <M>p</M>,
otherwise it is <E>fixed</E>.
Permutations in &GAP; are  entered and displayed in cycle notation,
such as <C>(1,2,3)(4,5)</C>.
<P/>
The preimage of the point <M>i</M> under the permutation <M>p</M> can be
computed as <M>i</M><C>/</C><M>p</M>, see <Ref Var="PERM_INVERSE_THRESHOLD"/>.
<P/>
For arithmetic operations for permutations and their precedence,
see <Ref Sect="Arithmetic Operations for Elements"/>.
<P/>
In the names of the &GAP; functions that deal with permutations,
the word <Q>Permutation</Q> is usually abbreviated to <Q>Perm</Q>,
to save typing.
For example,
the category test function for permutations is <Ref Filt="IsPerm"/>.


<Section Label="sect:IsPerm">
<Heading>IsPerm (Filter)</Heading>

<#Include Label="[1]{permutat}">
<#Include Label="IsPerm">
<#Include Label="IsPermCollection">
<#Include Label="PermutationsFamily">
<#Include Label="PERM_INVERSE_THRESHOLD">

</Section>


<Section Label="Comparison of Permutations">
<Heading>Comparison of Permutations</Heading>

<ManSection>
<Meth Name="\=" Arg='p1, p2' Label="for permutations"/>
<Meth Name="\<" Arg='p1, p2' Label="for permutations"/>

<Description>
<Index Subkey="for permutations">equality test</Index>
<Index Subkey="for permutations">precedence test</Index>
Two permutations are equal if they move the same points and all these points
have the same images under both permutations.
<P/>
The permutation <A>p1</A> is smaller than <A>p2</A>
if <A>p1</A> <M>\neq</M> <A>p2</A>
and <M>i^{{<A>p1</A>}} < i^{{<A>p2</A>}}</M>,
where <M>i</M> is the smallest point with
<M>i^{{<A>p1</A>}} \neq i^{{<A>p2</A>}}</M>.
Therefore the identity permutation is the smallest permutation,
see also Section <Ref Sect="Comparison Operations for Elements"/>.
<P/>
Permutations can be compared with certain other &GAP; objects,
see <Ref Sect="Comparisons"/> for the details.
<P/>
<Example><![CDATA[
gap> (1,2,3) = (2,3,1);
true
gap> (1,2,3) * (2,3,4) = (1,3)(2,4);
true
gap> (1,2,3) < (1,3,2);      # 1^(1,2,3) = 2 < 3 = 1^(1,3,2)
true
gap> (1,3,2,4) < (1,3,4,2);  # 2^(1,3,2,4) = 4 > 1 = 2^(1,3,4,2)
false
]]></Example>
</Description>
</ManSection>

<#Include Label="DistancePerms">
<#Include Label="SmallestGeneratorPerm">

</Section>


<Section Label="Moved Points of Permutations">
<Heading>Moved Points of Permutations</Heading>

<#Include Label="SmallestMovedPoint">
<#Include Label="LargestMovedPoint">
<#Include Label="MovedPoints">
<#Include Label="NrMovedPoints">

</Section>


<Section Label="Sign and Cycle Structure">
<Heading>Sign and Cycle Structure</Heading>

<#Include Label="SignPerm">
<#Include Label="CycleStructurePerm">

</Section>


<Section Label="Creating Permutations">
<Heading>Creating Permutations</Heading>

<#Include Label="ListPerm">
<#Include Label="PermList">
<#Include Label="MappingPermListList">
<#Include Label="RestrictedPerm">
<#Include Label="CycleFromList">

<ManSection>
  <Attr Name="AsPermutation" Arg="f"/>
  <Returns>A permutation or <K>fail</K>.</Returns>
  <Description>
    Partial permutations and transformations which define permutations
    (mathematically) can be converted into &GAP; permutations using
    <C>AsPermutation</C>; see Chapters <Ref Chap="Transformations"/> and
    <Ref Chap="Partial permutations"/> for more details about transformations
    and partial permutations.

    <List>
      <Mark>for partial permutations</Mark>
      <Item>
        If the partial permutation <A>f</A> is a permutation of its image, then
        <C>AsPermutation</C> returns this permutation.  If <A>f</A> does not
        permute its image, then <K>fail</K> is returned. <P/>
      </Item>

      <Mark>for transformations</Mark>
      <Item>
        A transformation is a permutation if and only if its rank equals its
        degree. If a transformation in &GAP; is a permutation, then
        <C>AsPermutation</C> returns this permutation.  If <A>f</A> is not a
        permutation, then <K>fail</K> is returned.
      </Item>
    </List>

    The function
    <Ref Func="Permutation" Label="for a group, an action domain, etc."/> can
    also be used to convert partial permutations and transformations into
    permutations where appropriate.
    <Example>
gap> f:=PartialPerm( [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ],
> [ 2, 7, 9, 4, 1, 10, 5, 6, 3, 8 ] );
(1,2,7,5)(3,9)(4)(6,10,8)
gap> AsPermutation(f);
(1,2,7,5)(3,9)(6,10,8)
gap> f:= PartialPerm( [ 1, 2, 3, 4, 5, 7, 8 ], [ 5, 3, 8, 1, 9, 4, 10 ] );
[2,3,8,10][7,4,1,5,9]
gap> AsPermutation(f);
fail
gap> f:=Transformation( [ 5, 8, 3, 5, 8, 6, 2, 2, 7, 8 ] );;
gap> AsPermutation(f);
fail
gap> f:=Transformation( [ 1, 3, 6, 6, 2, 10, 2, 3, 10, 5 ] );;
gap> AsPermutation(f);
fail
gap> f:=Transformation( [ 2, 7, 9, 4, 1, 10, 5, 6, 3, 8 ] );
Transformation( [ 2, 7, 9, 4, 1, 10, 5, 6, 3, 8 ] )
gap> AsPermutation(f);
(1,2,7,5)(3,9)(6,10,8)</Example>
    </Description>
  </ManSection>

</Section>
</Chapter>

quality93%

¤ Dauer der Verarbeitung: 0.24 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.