<!-- $Id: funct.xml,v 1.5 2007/03/14 20:41:57 alexk Exp $ -->
<Chapter Label="Funct">
<Heading>&UnitLib; functions</Heading>

Since the main purpose of &UnitLib; is the storage of large amount
of data, it has only two main user functions, which allow to read the 
description of <M>V(KG)</M> for the given catalogue number of <M>G</M> 
in the Small Groups Libary of the &GAP; system, and to save the description 
of <M>V(KG)</M> if the user would like to store it for further usage 
for the group that is not contained in the library.
<P/>

Examples below contain some functions from the &LAGUNA; package
<Cite Key="Laguna"/>, see their description in the &LAGUNA; manual.
<P/>

To use the &UnitLib; package first you need to load it as follows:

<Example>
<![CDATA[
gap> LoadPackage("unitlib");
----------------------------------------------------------------------------
Loading  UnitLib 2.1 (Library of normalized unit groups of modular group algebras)
by Alexander Konovalov (http://www.cs.st-andrews.ac.uk/~alexk/) and
   Elena Yakimenko (k-algebra@zsu.zp.ua).
----------------------------------------------------------------------------
true
gap>
]]>
</Example>

In case of a non-UNIX system, a warning will be displayed about non-availability
of the library of normalized unit groups for groups of orders 128 and 243. 
   
<Section Label="MainFunctions">
<Heading>MainFunctions</Heading>

<ManSection>
   <Func Name="PcNormalizedUnitGroupSmallGroup" 
         Arg="s n" />  
   <Returns>
         PcGroup
   </Returns>	 
   <Description>
         Let <A>s</A> be a power of prime <M>p</M> and
	 <A>n</A> is an integer from <C>[ 1 .. NrSmallGroups(s) ]</C>.
	 Then <C>PcNormalizedUnitGroupSmallGroup(<A>s</A>,<A>n</A>)</C> 
	 returns the normalized unit group <M>V(KG)</M> 
	 of the modular group algebra <M>KG</M>, where
	 <M>G</M> is <C>SmallGroup(<A>s</A>,<A>n</A>)</C>
	 and <M>K</M> is a field of <M>p</M> elements.
   </Description>
</ManSection>

<Example>
<![CDATA[
gap> PcNormalizedUnitGroupSmallGroup(128,161);
<pc group of size 170141183460469231731687303715884105728 with 127 generators>
]]>
</Example>

The result returned by <C>PcNormalizedUnitGroupSmallGroup</C>
will be equivalent to the following sequence of commands:
 
<Log>                                                                         
<![CDATA[                                                                         
gap> G := SmallGroup( s, n );
gap> p := PrimePGroup( G );
gap> K := GF( p );
gap> KG := GroupRing( K, G );
gap> PcNormalizedUnitGroup( KG );
]]>                                                                               
</Log> 

Nevertheless, <C>PcNormalizedUnitGroupSmallGroup</C>
is not just a shortcut for such computation. It reads the description
of the normalized unit group from the &UnitLib; library and then 
reconstructs all its necessary attributes and properties.
Thus, if you would like to obtain the group algebra <M>KG</M> or the
field <M>K</M> and the group <M>G</M>, you should extract them from
<M>V(KG)</M>, which should be constructed first.

<Example>
<![CDATA[
gap> V:=PcNormalizedUnitGroup(GroupRing(GF(2),SmallGroup(8,3)));
<pc group of size 128 with 7 generators>
gap> V1:=PcNormalizedUnitGroupSmallGroup(8,3);                   
<pc group of size 128 with 7 generators>
gap> V1=V;     # two isomorphic groups but not identical objects
false
gap> IdGroup(V)=IdGroup(V1);
true
gap> IsomorphismGroups(V,V1);
[ f1, f2, f3, f4, f5, f6, f7 ] -> [ f1, f2, f3, f4, f5, f6, f7 ]
gap> KG:=UnderlyingGroupRing(V1);  # now the correct way
<algebra-with-one over GF(2), with 3 generators>
gap> V1=PcNormalizedUnitGroup(KG); # V1 is an attribite of KG
true
gap> K:=UnderlyingField(KG);
GF(2)
gap> G:=UnderlyingGroup(KG);     
<pc group of size 8 with 3 generators>
]]>
</Example>

Moreover, the original group <M>G</M> can be embedded into the output of the
<C>PcNormalizedUnitGroupSmallGroup</C>, as it is shown in the continuation
of the previous example:

<Example>
<![CDATA[
gap> f:=Embedding(G,V1); 
[ f1, f2, f3 ] -> [ f2, f1, f3 ]
gap> g:=List(GeneratorsOfGroup(G), x -> x^f ); 
[ f2, f1, f3 ]
gap> G1:=Subgroup(V1,g);
Group([ f2, f1, f3 ])
gap> IdGroup(G1);
[ 8, 3 ]
]]>
</Example>

If the first argument <A>s</A> (the order of a group) is not a power of 
prime, an error message will appear.
If <A>s</A> is bigger than 243, you will get a warning telling that
the library does not contain <M>V(KG)</M> for <M>G</M> of such order, 
and you can use only data that you already stored in your 
<File>unitlib/userdata</File> directory with the help of the function
<Ref Func="SavePcNormalizedUnitGroup" />.
<P/>

It is worth to mention that for some groups of order 243, the construction 
of the normalized unit group using <C>PcNormalizedUnitGroupSmallGroup</C> 
may already require some noticeable amount of time. For example, it took 
about 166 seconds of CPU time to compute 
<C>PcNormalizedUnitGroupSmallGroup(243,30)</C> 
on Intel Xeon 3.4 GHz with 2048 KB cache.

<ManSection>
   <Prop Name="SavePcNormalizedUnitGroup" 
         Arg="G" />
   <Returns>
         true
   </Returns>	 
   <Description>
         Let <A>G</A> be a finite <M>p</M>-group of order <M>s</M>
	 from the Small Groups Library of the &GAP; system, 
	 constructed with the help of <C>SmallGroup(s,n)</C>. Then
	 <C>SavePcNormalizedUnitGroup(<A>G</A>)</C> 
         creates the file with the name of the form 
	 <File>us&uscore;n.g</File> in the directory 
	 <File>unitlib/userdata</File>, and returns <K>true</K>
	 if this file was successfully generated.
	 This file contains the description of the normalized unit 
	 group <M>V(KG)</M> of the group algebra of the group 
	 <A>G</A> over the field of <M>p</M> elements. 
	 <P/>
	 If the order of <A>G</A> is greater than 243, after this 
	 you can construct the group <M>V(KG)</M> using  
         <Ref Func="PcNormalizedUnitGroupSmallGroup" /> similarly
	 to the previous section.
	 The preliminary warning will be displayed, telling that
	 for such orders you can use only those groups that were
	 already computed by the user and saved to the 
	 <File>unitlib/userdata</File> directory. 
	 If there will be no such file there, you will get an error
	 message, otherwise the computation will begin.
	 <P/>
	 If the order of <A>G</A> is less or equal to 243, then
	 the file will be created in the <File>unitlib/userdata</File>
	 directory, but &UnitLib; will continue to use the file
	 with the same name from the appropriate directory in 
	 <File>unitlib/data</File>. You can compare these two files 
	 to make it sure that they are the same.
	 <P/>
	 <B>WARNINGS:</B> 
	 <P/>
	 1. It is important to apply this function to the
	 underlying group <M>G</M> and not to the normalized unit group
	 <M>V(KG)</M>.
	 <P/>
	 2. The user should use as an argument only 
	 groups from the Small Groups Library of the &GAP; system, 
	 constructed with the help of <C>SmallGroup(s,n)</C>, otherwise
	 the consistency of data may be lost.
   </Description>
</ManSection>

<Example>
<![CDATA[
gap> SavePcNormalizedUnitGroup( SmallGroup( 256, 56092 ) );
true
gap> PcNormalizedUnitGroupSmallGroup( 256, 56092 );
WARNING : the library of V(KG) for groups of order
256 is not available yet !!!
You can use only groups from the unitlib/userdata directory
in case if you already computed their descriptions
(See the manual for SavePcNormalizedUnitGroup)

Description of V(KG) for G=SmallGroup(256,
56092) accepted, started its generation
<pc group of size
57896044618658097711785492504343953926634992332820282019728792003956564819968
  with 255 generators>
]]>
</Example>

</Section>


<Alt Only="LaTeX">\newpage</Alt>


<Section Label="Service">
<Heading>Service tools</Heading>

<ManSection>
   <Func Name="UNITLIBBuildManual" 
         Arg=""
	 Comm="requires GAPDoc, UNIX or Linux and TeX" />
   <Description>
      This function is used to build the manual in the following formats:
      DVI, PDF, PS, HTML and text for online help.
      We recommend that the user should have a recent and fairly
      complete &TeX; distribution.
      Since &UnitLib; is distributed together with its manual,
      it is not necessary for the user to use this function. Normally
      it is intended to be used by the developers only. This is the only 
      function of &UnitLib; which requires UNIX/Linux environment.
   </Description>
</ManSection>


<ManSection>
   <Func Name="UNITLIBBuildManualHTML" 
         Arg=""
	 Comm="requires only GAPDoc" />
   <Description>
      This fuction is used to build the manual only in HTML format.
      This does not depend on the availability of the &TeX; installation
      and works under Windows and MacOS as well.
      Since &UnitLib; is distributed together with its manual,
      it is not necessary for the user to use this function. Normally
      it is intended to be used by the developers only.
   </Description>
</ManSection>

</Section>

</Chapter>