Quellcode-Bibliothek _Chapter_Twisted_conjugacy.xml Sprache: XML

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<Chapter Label="Chapter_Twisted_conjugacy">
<Heading>Twisted conjugacy</Heading>

<Section Label="Chapter_Twisted_conjugacy_Section_The_twisted_conjugation_action">
<Heading>The twisted conjugation action</Heading>

Let <Math>G</Math> and <Math>H</Math> be groups and let <Math>\varphi</Math> and <Math>\psi</Math> be group
homomorphisms from <Math>H</Math> to <Math>G</Math>. The pair <Math>(\varphi,\psi)</Math> induces a
(right) group action of <Math>H</Math> on <Math>G</Math> given by the map <Display>G \times H \to G
\colon (g,h) \mapsto \varphi(h)^{-1} g\,\psi(h).</Display>
This group action is called <Emph><Math>(\varphi,\psi)</Math>-twisted conjugation</Emph>.
<P/>
If <Math>G = H</Math>, <Math>\varphi</Math> is an endomorphism of <Math>G</Math> and
<Math>\psi = \operatorname{id}_G</Math>, then the action is usually called
<Emph><Math>\varphi</Math>-twisted conjugation</Emph>.
In general, for the &PACKAGENAME; package, many functions will take
two homomorphisms <A>hom1</A> and <A>hom2</A> as arguments. However, if
<A>hom1</A> is an endomorphism, <A>hom2</A> can be omitted, in which case it
is automatically taken to be the identity map.
<P/>
Similarly, some functions will take two elements <A>g1</A> and <A>g2</A> as
arguments. If <A>g2</A> is omitted, it is automatically taken to be the
identity element.
<ManSection>
  <Func Arg="hom1[, hom2]" Name="TwistedConjugation" />
<Returns>a function that maps the pair <C>(g,h)</C> to  <A>hom1</A><C>(h)⁻¹</C> <C>g</C> <A>hom2</A><C>(h)</C>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>

</Section>

<Section Label="Chapter_Twisted_conjugacy_Section_The_twisted_conjugacy_search_problem">
<Heading>The twisted conjugacy (search) problem</Heading>

Given groups <Math>G</Math> and <Math>H</Math>, group homomorphisms <Math>\varphi</Math> and <Math>\psi</Math> from <Math>H</Math>
to <Math>G</Math> and elements <Math>g_1, g_2 \in G</Math>, the <Emph>twisted conjugacy problem</Emph> is
the decision problem that asks whether <Math>g_1</Math> and <Math>g_2</Math> are
<Math>(\varphi,\psi)</Math>-twisted conjugate.
The <Emph>twisted conjugacy search problem</Emph> is the problem of determining
an explicit <Math>h</Math> such that <Math>\varphi(h)^{-1}g_1\psi(h) = g_2</Math> (under the
assumption that such <Math>h</Math> exists).
<ManSection>
  <Func Arg="hom1[, hom2], g1[, g2]" Name="IsTwistedConjugate" />
<Returns><K>true</K> if <A>g1</A> and <A>g2</A> are  <C>(<A>hom1</A>,<A>hom2</A>)</C>-twisted conjugate, otherwise <K>false</K>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>

<ManSection>
  <Func Arg="hom1[, hom2], g1[, g2]" Name="RepresentativeTwistedConjugation" />
<Returns>an element that maps <A>g1</A> to <A>g2</A> under the  <C>(<A>hom1</A>,<A>hom2</A>)</C>-twisted conjugacy action, or <K>fail</K> if
  no such element exists.
</Returns>
<Description>
If the source group is finite, this function relies on orbit-stabiliser
algorithms provided by &GAP;. Otherwise, it relies on a mixture of the
algorithms described in <Cite Key='roma16-a' Where='Thm. 3'/>,
<Cite Key='bkl20-a' Where='Sec. 5.4'/>,
<Cite Key='roma21-a' Where='Sec. 7'/> and <Cite Key='dt21-a'/>.
</Description>
</ManSection>

<Example><![CDATA[
gap> G := AlternatingGroup( 6 );;
gap> H := SymmetricGroup( 5 );;
gap> phi := GroupHomomorphismByImages( H, G, [ (1,2)(3,5,4), (2,3)(4,5) ],
>  [ (1,4)(3,6), () ] );;
gap> psi := GroupHomomorphismByImages( H, G, [ (1,2)(3,5,4), (2,3)(4,5) ],
>  [ (1,2)(3,4), () ] );;
gap> tc := TwistedConjugation( phi, psi );;
gap> g1 := (4,6,5);;
gap> g2 := (1,6,4,2)(3,5);;
gap> IsTwistedConjugate( psi, phi, g1, g2 );
false
gap> h := RepresentativeTwistedConjugation( phi, psi, g1, g2 );
(1,2)
gap> tc( g1, h ) = g2;
true
]]></Example>

</Section>

<Section Label="Chapter_Twisted_conjugacy_Section_The_multiple_twisted_conjugacy_search_problem">
<Heading>The multiple twisted conjugacy (search) problem</Heading>

Let <Math>H</Math> and <Math>G_1, \ldots, G_n</Math> be groups. For each <Math>i \in \{1,\ldots,n\}</Math>,
let <Math>g_i,g_i' \in G_i and let

\varphi_i,\psi_i\colon H \to G_i

be group
homomorphisms.
The <Emph>multiple twisted conjugacy problem</Emph> is
the decision problem that asks whether there exists some <Math>h \in H</Math> such that
<Math>\varphi_i(h)^{-1}g_i\psi_i(h) = g_i' for all

i \in \{1,\ldots,n\}

.
The <Emph>multiple twisted conjugacy search problem</Emph> is the problem of
determining an explicit <Math>h</Math> such that <Math>\varphi_i(h)^{-1}g_i\psi_i(h) = g_i'
for all <Math>i \in \{1,\ldots,n\}</Math> (under the assumption that such <Math>h</Math> exists).
<P/>
<Ref Func="IsTwistedConjugate" Style="Number"/> and
<Ref Func="RepresentativeTwistedConjugation" Style="Number"/> can take lists
instead of their usual arguments to solve these problems.
<Example><![CDATA[
gap> H := SymmetricGroup( 5 );;
gap> G := AlternatingGroup( 6 );;
gap> phi := GroupHomomorphismByImages( H, G, [ (1,2)(3,5,4), (2,3)(4,5) ],
>  [ (1,4)(3,6), () ] );;
gap> psi := GroupHomomorphismByImages( H, G, [ (1,2)(3,5,4), (2,3)(4,5) ],
>  [ (1,2)(3,4), () ] );;
gap> tau := GroupHomomorphismByImages( H, G, [ (1,2)(3,5,4), (2,3)(4,5) ],
>  [ (1,2)(3,6), () ] );;
gap> khi := GroupHomomorphismByImages( H, G, [ (1,2)(3,5,4), (2,3)(4,5) ],
>  [ (1,3)(4,6), () ] );;
gap> IsTwistedConjugate( [ phi, psi ], [ khi, tau ],
>  [ (1,5)(4,6), (1,4)(3,5) ], [ (1,4,5,3,6), (2,4,5,6,3) ] );
true
gap> RepresentativeTwistedConjugation( [ phi, psi ], [ khi, tau ],
>  [ (1,5)(4,6), (1,4)(3,5) ], [ (1,4,5,3,6), (2,4,5,6,3) ] );
(1,2)
]]></Example>

</Section>

</Chapter>

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