<!-- This is an automatically generated file. -->
<Chapter Label="Chapter_Affine_actions">
<Heading>Affine actions</Heading>
Let <Math>G</Math> and <Math>H</Math> be groups, let <Math>H</Math> act on <Math>G</Math> (via automorphisms) by
<Display>\alpha \colon H \to \operatorname{Aut}(G) \colon h \mapsto \alpha_h</Display>
and let <Math>\delta \colon H \to G</Math> be a group derivation with respect to this
action. Then we can construct a new action, called the <Emph>affine action</Emph>
associated to <Math>\delta</Math>, by
<Display>G \times H \to G \colon g^h = \alpha_h(g) \delta(h).</Display>
If <Math>K</Math> is a subgroup of <Math>H</Math>, then the restriction of the affine action of
<Math>H</Math> on <Math>G</Math> to <Math>K</Math> coincides with the affine action of <Math>K</Math> on <Math>G</Math> associated
to the restriction of <Math>\delta</Math> to <Math>K</Math>.
<P/>
Algorithms designed for computing with twisted conjugacy classes can be
leveraged to do computations involving affine actions, see
<Cite Key='tert25-a' Where='Sec. 10'/> for a description on this.
<P/>
Please note that the functions in this chapter require <Math>G</Math> and <Math>H</Math> to either
both be finite, or both be PcpGroups.
<Section Label="Chapter_Affine_actions_Section_Creating_an_affine_action">
<Heading>Creating an affine action</Heading>
<ManSection Label="AutoDoc_generated_group32">
<Func Arg="K, der" Name="AffineActionByGroupDerivation" />
<Returns>the affine action of <A>K</A> associated to the derivation <A>der</A>.
</Returns>
<Description>
The group <A>K</A> must be a subgroup of <C>Source(<A>der</A>)</C>.
</Description>
</ManSection>
<Example><![CDATA[
gap> aff := AffineActionByGroupDerivation( H, der );
function( g, k ) ... end
]]></Example>
</Section>
<Section Label="Chapter_Affine_actions_Section_Operations_for_affine_actions">
<Heading>Operations for affine actions</Heading>
These functions are analogues of existing &GAP; functions for group actions.
<ManSection Label="AutoDoc_generated_group33">
<Heading>OrbitAffineAction</Heading>
<Func Arg="K, g, der" Name="OrbitAffineAction" />
<Returns>the orbit of <A>g</A> under the affine action of <A>K</A> associated to <A>der</A>.
</Returns>
<Description>
The group <A>K</A> must be a subgroup of <C>Source(<A>der</A>)</C>.
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group34">
<Heading>OrbitsAffineAction</Heading>
<Func Arg="K, der" Name="OrbitsAffineAction" />
<Returns>a list containing the orbits under the affine action of <A>K</A> associated to <A>der</A> if there are finitely many, or <K>fail</K> if there
are infinitely many.
</Returns>
<Description>
The group <A>K</A> must be a subgroup of <C>Source(<A>der</A>)</C>.
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group35">
<Heading>NrOrbitsAffineAction</Heading>
<Func Arg="K, der" Name="NrOrbitsAffineAction" />
<Returns>the number of orbits under the affine action of <A>K</A> associated to <A>der</A>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group36">
<Heading>StabiliserAffineAction</Heading>
<Func Arg="K, g, der" Name="StabiliserAffineAction" />
<Func Arg="K, g, der" Name="StabilizerAffineAction" />
<Returns>the stabiliser of <A>g</A> under the affine action of <A>K</A> associated to <A>der</A>.
</Returns>
<Description>
The group <A>K</A> must be a subgroup of <C>Source(<A>der</A>)</C>.
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group37">
<Heading>RepresentativeAffineAction</Heading>
<Func Arg="K, g1, g2, der" Name="RepresentativeAffineAction" />
<Returns>an element of <A>K</A> that maps <A>g1</A> to <A>g2</A> under the affine action of <A>K</A> associated to <A>der</A>, or <K>fail</K> if no such element exists.
</Returns>
<Description>
The group <A>K</A> must be a subgroup of <C>Source(<A>der</A>)</C>.
</Description>
</ManSection>
<Example><![CDATA[
gap> g1 := G.1;;
gap> orb := OrbitAffineAction( H, g1, der );
f1^G
gap> NrOrbitsAffineAction( H, der );
10
gap> stab := StabiliserAffineAction( H, g1, der );;
gap> Set( stab );
[ <identity> of ..., f3, f3^2, f2^2*f5, f2*f4*f5,
f2^2*f3*f5, f2*f3*f4*f5, f2^2*f3^2*f5, f2*f3^2*f4*f5 ]
gap> g2 := G.1*G.4*G.5;;
gap> h := RepresentativeAffineAction( H, g1, g2, der );;
gap> aff( g1, h ) = g2;
true
]]></Example>
</Section>
<Section Label="Chapter_Affine_actions_Section_Operations_on_orbits_of_affine_actions">
<Heading>Operations on orbits of affine actions</Heading>
<ManSection Label="AutoDoc_generated_group38">
<Heading>Representative</Heading>
<Attr Arg="orb" Name="Representative" Label="of an orbit of an affine action"/>
<Returns>the group element that was used to construct <A>orb</A>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group39">
<Heading>ActingDomain</Heading>
<Attr Arg="orb" Name="ActingDomain" Label="of an orbit of an affine action"/>
<Returns>the group whose affine action <A>orb</A> is an orbit of.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group40">
<Heading>FunctionAction</Heading>
<Attr Arg="orb" Name="FunctionAction" Label="of an orbit of an affine action"/>
<Returns>the affine action that <A>orb</A> is an orbit of.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group41">
<Heading>\in</Heading>
<Oper Arg="elm, orb" Name="\in" Label="for an element and an orbit of an affine action"/>
<Returns><K>true</K> if <A>elm</A> is an element of <A>orb</A>, otherwise <K>false</K>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group42">
<Heading>Size</Heading>
<Attr Arg="orb" Name="Size" Label="of an orbit of an affine action"/>
<Returns>the number of elements in <A>orb</A>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group43">
<Heading>StabiliserOfExternalSet</Heading>
<Attr Arg="orb" Name="StabiliserOfExternalSet" Label="of an orbit of an affine action"/>
<Returns>the stabiliser of <C>Representative(<A>orb</A>)</C> under the action <C>FunctionAction(<A>orb</A>)</C>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group44">
<Heading>List</Heading>
<Func Arg="orb" Name="List" Label="of an orbit of an affine action"/>
<Returns>a list containing the elements of <A>orb</A>.
</Returns>
<Description>
If <A>orb</A> is infinite, this will run forever. It is recommended to first
test the finiteness of <A>orb</A> using
<Ref Attr="Size" Label="of an orbit of an affine action" Style="Number"/>.
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group45">
<Heading>Random</Heading>
<Oper Arg="orb" Name="Random" Label="in an orbit of an affine action"/>
<Returns>a random element in <A>orb</A>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group46">
<Heading>\=</Heading>
<Oper Arg="orb1, orb2" Name="\=" Label="for orbits of an affine action"/>
<Returns><K>true</K> if <A>orb1</A> is equal to <A>orb2</A>, otherwise <K>false</K>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<Example><![CDATA[
gap> g2 in orb;
true
gap> G.2 in orb;
false
gap> Size( orb );
8
]]></Example>
</Section>
</Chapter>
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