Quelle _Chapter_The_SOTGrps_package.xml Sprache: XML

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<Chapter Label="Chapter_The_SOTGrps_package">
<Heading>The &SOTGrps; package</Heading>

With some overlaps, the &SOTGrps; package extends the Small Group Library to give access to some more
    <Q>small</Q> orders. For example, it constructs a
    complete and irredundant list of isomorphism type representatives of the groups of order
<List>
<Item>
that factorises into at most four primes;
</Item>
<Item>
<M>p^4q</M>, for distinct primes <M>p</M> and <M>q</M>.
</Item>
</List>
<P/>
The mathematical background for this package is described in <Cite Key="DEP22"/>.
<Section Label="Chapter_The_SOTGrps_package_Section_Main_functions">
<Heading>Main functions</Heading>

<P/>
In addition to the functions described below, the &SOTGrps; package also extends the
the Small Groups Library as provided by the &SmallGrp; package: with &SOTGrps; loaded,
functions such as <C>NumberSmallGroups</C>, <C>SmallGroup</C> or <C>IdGroup</C>
will work for orders support by &SOTGrps; but not by &SmallGrp;.
<P/>
Note: for orders support by &SOTGrps; *and* by &SmallGrp;, the respective ids as
produced by <C>IdGroup</C> versus <C>IdSOTGroup</C> in general do not agree.
In a future version we may provided functions to convert between them.
<ManSection>
  <Func Arg="n [, filter]" Name="AllSOTGroups" />
<Description>
  takes in a number <A>n</A> that factorises into at most four primes or is of the form <M>p^4q</M> (<M>p</M>, <M>q</M> are distinct primes),
  and returns a complete and duplicate-free list of isomorphism class representatives of the groups of order <A>n</A>.
  Solvable groups are using refined polycyclic presentations.
  By default, solvable groups are constructed in the filter <C>IsPcGroup</C>,
  but if the optional argument <A>filter</A> is set to <C>IsPcpGroup</C> then
  the groups are constructed in that filter instead.
  Nonsolvable groups are always returned as permutation groups.
<Example><![CDATA[
gap> AllSOTGroups(60);
[ <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
  <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
  <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
  <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
  <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
  <pc group of size 60 with 4 generators>, <pc group of size 60 with 4 generators>,
Alt( [ 1 .. 5 ] ) ]
]]></Example>

</Description>
</ManSection>

<ManSection>
  <Func Arg="n" Name="NumberOfSOTGroups" />
<Description>
  takes in a number <A>n</A> that factorises into at most four primes or of the form <M>p^4q</M> (<M>p</M>, <M>q</M> are distinct primes),
  and returns the number of isomorphism types of groups of order <A>n</A>.
<Example><![CDATA[
gap> NumberOfSOTGroups(2*3*5*7);
12
gap> NumberOfSOTGroups(2*3*5*7*11);
Error, Order 2310 is not supported by SOTGrps.
Please refer to the SOTGrps documentation for the list of supported orders.
]]></Example>

</Description>
</ManSection>

<ManSection>
  <Func Arg="n, i[, arg]" Name="SOTGroup" />
<Description>
  takes in a pair of numbers <A>n, i</A>, where <A>n</A> factorises into at most four primes or of the form <M>p^4q</M> (<M>p</M>, <M>q</M> are distinct primes),
  and returns the <A>i</A>-th group with respect to the ordering of
  the list <C>AllSOTGroups(<A>n</A>)</C> without constructing all groups in the list.
  The option of constructing a PcpGroup is available for solvable groups.
<Example><![CDATA[
gap> SOTGroup(2*3*5*7, 1);
<pc group of size 210 with 4 generators>
]]></Example>

  If the input <A>i</A> exceeds the number of groups of order <A>n</A>, an error message is returned.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="G" Name="IdSOTGroup" Label="for IsGroup"/>
<Description>
  takes in a group of order determines the SOT library number of <A>G</A>;
  that is, the function returns a pair [<A>n</A>, <A>i</A>] where <A>G</A> is isomorphic to <C>SOTGroup(<A>n</A>,<A>i</A>)</C>.
  Note that if the input group is a PcpGroup, this may result in slow runtime, as <C>IdSOTGroup</C> may compute the <C>Centre</C> and/or the <C>FittingSubgroup</C>,
  which is slow for PcpGroups.
</Description>
</ManSection>

<ManSection>
  <Func Arg="G, H" Name="IsIsomorphicSOTGroups" />
<Description>
determines whether two groups <A>G</A>, <A>H</A> are isomorphic. It is assumed that the input groups are available in the &SOTGrps; library.
<Example><![CDATA[
gap> G:=Image(IsomorphismPermGroup(SmallGroup(690,1)));;
gap> H:=Image(IsomorphismPcGroup(SmallGroup(690,1)));;
gap> IsIsomorphicSOTGroups(G,H);
true
]]></Example>

</Description>
</ManSection>

<ManSection>
  <Func Arg="n" Name="IsSOTAvailable" />
<Description>
  returns <K>true</K> if the order <A>n</A> is available in the &SOTGrps; library, and <K>false</K> otherwise.
</Description>
</ManSection>

<ManSection>
  <Func Arg="n" Name="SOTGroupsInformation" />
<Description>
  prints information on the groups of the specified order.
  Since there are some overlaps between the existing SmallGrps library and the &SOTGrps; library.
  In particular, &SOTGrps; may construct the groups in a different order and so generate a different group ID; we denote such IDs by <K>SOT</K>.
  If the order covered in &SOTGrps; library has no conflicts with the existing library, then such a flag is removed.
<Example><![CDATA[
gap> SOTGroupsInformation(2^2*3*19);

  There are 15 groups of order 228.

  The groups of order p^2qr are either solvable or isomorphic to Alt(5).
  The solvable groups are sorted by their Fitting subgroup.
     SOT 1 - 2 are the nilpotent groups.
     SOT 3 has Fitting subgroup of order 57.
     SOT 4 - 7 have Fitting subgroup of order 76.
     SOT 8 - 9 have Fitting subgroup of order 38.
     SOT 10 - 15 have Fitting subgroup of order 114.

gap> SOTGroupsInformation(2662);

There are 15 groups of order 2662.

The groups of order p^3q are solvable by Burnside's pq-Theorem.
These groups are sorted by their Sylow subgroups.
   1 - 3 are abelian.
   4 - 5 are nonabelian nilpotent and have a normal Sylow 11-subgroup and a
       normal Sylow 2-subgroup.
   6 is non-nilpotent and has a normal Sylow 2-subgroup with Sylow
      11-subgroup [ 1331, 1 ].
   7 - 9 are non-nilpotent and have a normal Sylow 2-subgroup with Sylow
      11-subgroup [ 1331, 2 ].
   10 - 12 are non-nilpotent and have a normal Sylow 2-subgroup with Sylow
      11-subgroup [ 1331, 5 ].
   13 - 14 are non-nilpotent and have a normal Sylow 2-subgroup with Sylow
      11-subgroup [ 1331, 3 ].
   15 is non-nilpotent and has a normal Sylow 2-subgroup with Sylow
     11-subgroup [ 1331, 4 ].
]]></Example>

</Description>
</ManSection>

</Section>

</Chapter>

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