Quelle lins_interface.xml Sprache: XML

<Chapter Label="LINS Interface">
<Heading>LINS Interface</Heading>

This chapter is intended for advanced users.
It explains the provided search methods
and the interface to the search graph structure <C>LinsGraph</C>.

<Section Label="LINS Graph">
<Heading>LINS Graph</Heading>

All search methods in &LINS; return a <C>LinsGraph</C>
encoding a partial normal subgroup lattice of a finitely presented group <M>G</M>.

A <C>LinsGraph</C> is a graph, where each node is a <C>LinsNode</C> that contains a normal subgroup <M>H</M> of <M>G</M>
and pointers to the minimal <M>G</M>-normal super/sub-groups of <M>H</M>, i.e. its neighbours in the graph.
The directed edges of the graph are therefore encoded directly into the nodes.

<ManSection>
<Meth Name="List" Arg="gr" Label="for a lins graph"/>
<Description>
Returns a list of all <C>LinsNodes</C> in the graph <A>gr</A>. <P/>

The nodes are sorted by index in increasing order,
e.g. the root node is at the first position.

In order to get a list containing only the normal subgroups
that the search graph attempted to find, use <Ref Func="ComputedNormalSubgroups"/>. <P/>
</Description>
</ManSection>

<#Include Label="ComputedNormalSubgroups">
<#Include Label="LinsRoot">
<#Include Label="IndexBound">
<#Include Label="LinsOptions">

<ManSection>
<Attr Name="IsomorphismFpGroup" Arg="gr" Label="for a lins graph"/>
<Description>
Returns the isomorphism from the original group of the search
onto the fp-group contained in the root.
</Description>
</ManSection>

</Section>

<Section Label="LINS Node">
<Heading>LINS Node</Heading>

A <C>LinsNode</C> is a part of the search graph structure <C>LinsGraph</C> (see <Ref Sect="LINS Graph"/>).
As such, all methods are with respect to the search graph, where the node is contained in.

<ManSection>
<Meth Name="Grp" Arg="rH" Label="for a lins node"/>
<Description>
Returns the group contained in the node.
</Description>
</ManSection>

<ManSection>
<Meth Name="Index" Arg="rH" Label="for a lins node"/>
<Description>
Let <M>G</M> be the group contained in the root node
and <M>H</M> be the <M>G</M>-normal subgroup contained in <A>rH</A>. <P/>
Returns the index <M>[G : H]</M>.
</Description>
</ManSection>

<#Include Label="LinsNodeMinimalSupergroups">
<#Include Label="LinsNodeMinimalSubgroups">
<#Include Label="LinsNodeSupergroups">
<#Include Label="LinsNodeSubgroups">

</Section>

<Section Label="LINS Search Functions">
<Heading>LINS Search Functions</Heading>

<#Include Label="LowIndexNormalSubgroupsSearch">
<#Include Label="LowIndexNormalSubgroupsSearchForAll">
<#Include Label="LowIndexNormalSubgroupsSearchForIndex">

</Section>

<Section Label="Advanced Examples">
<Heading>Examples</Heading>

In this section we present example sessions which demonstrate
how to use the advanced search methods provided by &LINS;.
For this we revise the examples from the introduction as well as include new ones. <P/>

<Subsection>
<Heading>Revised Example : all normal subgroups up to index <M>n</M></Heading>

We compute all normal subgroups in <M>D_{50}</M>,
the dihedral group of size <M>50</M>.
<Example><![CDATA[
gap> G := DihedralGroup(50);
<pc group of size 50 with 3 generators>
]]></Example>

The search algorithm automatically translates the group into a finitely presented group
via a call to <C>IsomorphismFpGroup</C>. <Br/>
The isomorphism is stored inside the lins graph.
<Example><![CDATA[
gap> gr := LowIndexNormalSubgroupsSearchForAll(G, 50);
<lins graph contains 4 normal subgroups up to index 50>
gap> r := LinsRoot(gr);
<lins node of index 1>
gap> H := Grp(r);
<fp group of size 50 on the generators [ F1, F2, F3 ]>
gap> Iso := IsomorphismFpGroup(gr);
[ f1, f2, f3 ] -> [ F1, F2, F3 ]
gap> Source(Iso) = G;
true
gap> Range(Iso) = H;
true
]]></Example>

In order to get all nodes from the search graph, we need to use <C>List</C>.
As expected, the algorithm finds <M>D_{50}, C_{25}, C_5</M>
and the trivial group.
<Example><![CDATA[
gap> L := List(gr);
[ <lins node of index 1>, <lins node of index 2>, <lins node of index 10>,
<lins node of index 50> ]
gap> IsoTypes := List(L, node -> StructureDescription(Grp(node)));
[ "D50", "C25", "C5", "1" ]
]]></Example>

</Subsection>

<Subsection>
<Heading>Revised Example : all normal subgroups of index <M>n</M></Heading>

We compute all normal subgroups of index <M>5^2 = 25</M> in <M>C_5^4</M>,
the direct product of <M>4</M> copies of the cyclic group of order <M>5</M>:
<Example><![CDATA[
gap> G := ElementaryAbelianGroup(5^4);
<pc group of size 625 with 4 generators>
]]></Example>

Again, the search algorithm automatically translates the group into a finitely presented group
via a call to <C>IsomorphismFpGroup</C>.
<Example><![CDATA[
gap> gr := LowIndexNormalSubgroupsSearchForIndex(G, 5 ^ 2, infinity);
<lins graph contains 963 normal subgroups up to index 25>
]]></Example>

Now we are not interested in all normal subgroups that the search graph considered,
but only in those of index <M>25</M>. Thus we need to use <C>ComputedNormalSubgroups</C>.
For a prime <M>p</M>, and integers <M>d, s \in \mathbb{N}</M>,
the number of subgroups of order <M>p^s</M> of an elementary abelian <M>p</M>-group of order <M>p^d</M>
is exactly
<Alt Not="Text,HTML"><Display>
\frac
{\left(p^d - 1\right)\left(p^d - p\right) \cdots \left(p^d - p^{s-1}\right)}
{\left(p^s - 1\right)\left(p^s - p\right) \cdots \left(p^s - p^{s-1}\right)}\;.
</Display></Alt>
<Alt Only="Text,HTML"><Display Mode="M">
( (p^d - 1)(p^d - p) \cdots (p^d - p^{(s-1)}) ) / ( (p^s - 1)(p^s - p) \cdots (p^s - p^{(s-1)}) ) .
</Display></Alt>
Thus we expect to find
<Alt Not="Text,HTML"><M>\frac{(5^4-1) \cdot (5^4-5)}{(5^2 - 1) \cdot (5^2 - 5)} = 806</M></Alt>
<Alt Only="Text,HTML"><M>( (5^4-1) \cdot (5^4-5) ) / ( (5^2 - 1) \cdot (5^2 - 5) ) = 806</M></Alt>
normal subgroups of index <M>25</M>. <Br/>
Furthermore, all subgroups need to be of the isomorphism type <M>C_5^2</M>.
<Example><![CDATA[
gap> L := ComputedNormalSubgroups(gr);;
gap> IsoTypes := Collected(List(L, node -> StructureDescription(Grp(node))));
[ [ "C5 x C5", 806 ] ]
]]></Example>

</Subsection>

<Subsection>
<Heading>Example : a normal subgroup of index <M>n</M></Heading>

We compute a normal subgroup of index <M>3 \cdot 5 = 15</M> in
<M>C_3 \times C_3 \times C_4 \times C_5</M>,
a direct product of cyclic groups:
<Example><![CDATA[
gap> G := AbelianGroup([3, 3, 4, 5]);
<pc group of size 180 with 4 generators>
gap> gr := LowIndexNormalSubgroupsSearchForIndex(G, 15, 1);
<lins graph contains 7 normal subgroups up to index 15>
]]></Example>

We use <C>ComputedNormalSubgroups</C> in order to get the normal subgroup of index <M>15</M>.
As expected, the algorithm finds a group of the isomorphism type <M>C_{12} = C_3 \times C_4</M>.
<Example><![CDATA[
gap> L := ComputedNormalSubgroups(gr);
[ <lins node of index 15> ]
gap> IsoTypes := List(L, node -> StructureDescription(Grp(node)));
[ "C12" ]
]]></Example>

</Subsection>

</Section>

</Chapter>

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