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<?xmlversion="1.0"encoding="UTF-8"?>
<!-- This is an automatically generated file. -->
<Chapter Label="Chapter_Group_derivations">
<Heading>Group derivations</Heading>
Let <Math>G</Math> and <Math>H</Math> be groups and let <Math>H</Math> act on <Math>G</Math> via automorphisms, i.e.
there is a group homomorphism
<Display>\alpha \colon H \to \operatorname{Aut}(G) \colon h \mapsto \alpha_h</Display>
such that <Math>g^h = \alpha_h(g)</Math> for all <Math>g \in G</Math> and <Math>h \in H</Math>.
A <Emph>group derivation</Emph> <Math>\delta \colon H \to G</Math> is a map such that
<Display>\delta(h_1h_2) = \delta(h_1)^{h_2}\delta(h_2).</Display>
Note that we do not require <Math>G</Math> to be abelian.
<P/>
Algorithms designed for computing with twisted conjugacy classes can be
leveraged to do computations involving group derivations, 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_Group_derivations_Section_Creating_group_derivations">
<Heading>Creating group derivations</Heading>
<ManSection Label="AutoDoc_generated_group21">
<Func Arg="H, G[[, gens], imgs], act" Name="GroupDerivationByImages" />
<Func Arg="H, G[[, gens], imgs], act" Name="GroupDerivationByImagesNC" />
<Returns>the specified group derivation, or <K>fail</K> if the given arguments do not define a derivation.
</Returns>
<Description>
This works in the same vein as
<Ref Func="GroupHomomorphismByImages" BookName="Ref" Style="Number"/>. The
group <A>H</A> acts on the group <A>G</A> via <A>act</A>, which must be a
homomorphism from <A>H</A> into a group of automorphisms of <A>G</A>. This
command then returns the group derivation defined by mapping the list
<A>gens</A> of generators of <A>H</A> to the list <A>imgs</A> of images in
<A>G</A>.
<P/>
If omitted, the arguments <A>gens</A> and <A>imgs</A> default to the
<C>GeneratorsOfGroup</C> value of <A>H</A> and <A>G</A> respectively.
<P/>
This function checks whether <A>gens</A> generate <A>H</A> and whether the
mapping of the generators extends to a group derivation. This test can be
expensive, so if one is certain that the given arguments produce a group
derivation, these checks can be avoided by using the <C>NC</C> version.
<P/>
</Description>
</ManSection>
<ManSection>
<Func Arg="H, G, fun, act" Name="GroupDerivationByFunction" />
<Returns>the specified group derivation.
</Returns>
<Description>
<C>GroupDerivationByFunction</C> works in the same vein as
<Ref Func="GroupHomomorphismByFunction" BookName="Ref" Style="Number"/>. The
group <A>H</A> acts on the group <A>G</A> via <A>act</A>, which must be a
homomorphism from <A>H</A> into a group of automorphisms of <A>G</A>. This
command then returns the group derivation defined by mapping the element
<C>h</C> of <A>H</A> to the element <A>fun</A>( <C>h</C> ) of <A>G</A>,
where <A>fun</A> is a &GAP; function.
<P/>
No tests are performed to check whether the arguments really produce a group
derivation.
</Description>
</ManSection>
<ManSection>
<Func Arg="H, G, act" Name="GroupDerivationByAffineAction" />
<Returns>the derivation that makes up the translational part of the affine action.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<Section Label="Chapter_Group_derivations_Section_Operations_for_group_derivations">
<Heading>Operations for group derivations</Heading>
Many of the functions, operations, attributes... available to group
homomorphisms are available for group derivations as well.
We list some of the more useful ones.
<ManSection Label="AutoDoc_generated_group22">
<Heading>IsInjective</Heading>
<Prop Arg="der" Name="IsInjective" Label="for a group derivation"/>
<Returns><K>true</K> if the group derivation <A>der</A> is injective, otherwise <K>false</K>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group23">
<Heading>IsSurjective</Heading>
<Prop Arg="der" Name="IsSurjective" Label="for a group derivation"/>
<Returns><K>true</K> if the group derivation <A>der</A> is surjective, otherwise <K>false</K>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group24">
<Heading>IsBijective</Heading>
<Prop Arg="der" Name="IsBijective" Label="for a group derivation"/>
<Returns><K>true</K> if the group derivation <A>der</A> is bijjective, otherwise <K>false</K>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group25">
<Heading>Kernel</Heading>
<Oper Arg="der" Name="Kernel" Label="of a group derivation"/>
<Returns>the set of elements that are mapped to the identity by <A>der</A>.
</Returns>
<Description>
This will always be a subgroup of <C>Source</C>(<A>der</A>).
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group26">
<Heading>Image</Heading>
<Func Arg="der" Name="Image" Label="of a group derivation"/>
<Func Arg="der, elm" Name="Image" Label="of an element under a group derivation"/>
<Func Arg="der, sub" Name="Image" Label="of a subgroup under a group derivation"/>
<Returns>the image of the group derivation <A>der</A>.
</Returns>
<Description>
One can optionally give an element <A>elm</A> or a subgroup <A>sub</A> as a
second argument, in which case <C>Image</C> will calculate the image of this
argument under <A>der</A>.
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group27">
<Heading>PreImagesRepresentative</Heading>
<Oper Arg="der, elm" Name="PreImagesRepresentative" Label="of an element under a group derivation"/>
<Returns>a preimage of the element <A>elm</A> under the group derivation <A>der</A>, or <K>fail</K> if no preimage exists.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group28">
<Heading>PreImages</Heading>
<Func Arg="der, elm" Name="PreImages" Label="of an element under a group derivation"/>
<Returns>the set of all preimages of the element <A>elm</A> under the group derivation <A>der</A>.
</Returns>
<Description>
This will always be a (right) coset of <C>Kernel</C>( <A>der</A> ), or the
empty list.
</Description>
</ManSection>
<Example><![CDATA[
gap> IsInjective( der ) or IsSurjective( der );
false
gap> K := Kernel( der );;
gap> Size( K );
9
gap> ImH := Image( der );
Group derivation image in Group( [ f1, f2, f3, f4, f5 ] )
gap> h1 := H.1*H.3;;
gap> g := Image( der, h1 );
f2*f4
gap> ImK := Image( der, K );
Group derivation image in Group( [ f1, f2, f3, f4, f5 ] )
gap> h2 := PreImagesRepresentative( der, g );;
gap> Image( der, h2 ) = g;
true
gap> PreIm := PreImages( der, g );
RightCoset(<group of size 9 with 2 generators>,<object>)
gap> PreIm = RightCoset( K, h2 );
true
]]></Example>
</Section>
<Section Label="Chapter_Group_derivations_Section_Images_of_group_derivations">
<Heading>Images of group derivations</Heading>
In general, the image of a group derivation is not a subgroup. However, it
is still possible to do a membership test, to calculate the number of
elements, and to enumerate the elements if there are only finitely many.
<ManSection Label="AutoDoc_generated_group29">
<Heading>\in</Heading>
<Oper Arg="elm, img" Name="\in" Label="for an element and a group derivation"/>
<Returns><K>true</K> if <A>elm</A> is an element of <A>img</A>, otherwise <K>false</K>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group30">
<Heading>Size</Heading>
<Attr Arg="img" Name="Size" Label="of a group derivation image"/>
<Returns>the number of elements in <A>img</A>.
</Returns>
<Description>
<P/>
</Description>
</ManSection>
<ManSection Label="AutoDoc_generated_group31">
<Heading>List</Heading>
<Func Arg="img" Name="List" Label="of a group derivation image"/>
<Returns>a list containing the elements of <A>img</A>.
</Returns>
<Description>
If <A>img</A> is infinite, this will run forever. It is recommended to first
test the finiteness of <A>img</A> using
<Ref Attr="Size" Label="of a group derivation image" Style="Number"/>.
</Description>
</ManSection>
<Example><![CDATA[
gap> Size( ImH );
8
gap> Size( ImK );
1
gap> g in ImH;
true
gap> g in ImK;
false
gap> List( ImK );
[ <identity> of ... ]
]]></Example>
</Section>
</Chapter>
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