&UnitLib; functions

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 V(KG) for the given catalogue number of G in the Small Groups Libary of the &GAP; system, and to save the description of V(KG) if the user would like to store it for further usage for the group that is not contained in the library.

Examples below contain some functions from the &LAGUNA; package , see their description in the &LAGUNA; manual.

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

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>

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.

MainFunctions
PcGroup
Let s be a power of prime p and n is an integer from [ 1 .. NrSmallGroups(s) ]. Then PcNormalizedUnitGroupSmallGroup(s,n) returns the normalized unit group V(KG) of the modular group algebra KG, where G is SmallGroup(s,n) and K is a field of p elements.
gap> PcNormalizedUnitGroupSmallGroup(128,161); <pc group of size 170141183460469231731687303715884105728 with 127 generators>

The result returned by PcNormalizedUnitGroupSmallGroup will be equivalent to the following sequence of commands:

gap> G := SmallGroup( s, n ); gap> p := PrimePGroup( G ); gap> K := GF( p ); gap> KG := GroupRing( K, G ); gap> PcNormalizedUnitGroup( KG );

Nevertheless, PcNormalizedUnitGroupSmallGroup 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 KG or the field K and the group G, you should extract them from V(KG), which should be constructed first.

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>

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

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 ]

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

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

true
Let G be a finite p-group of order s from the Small Groups Library of the &GAP; system, constructed with the help of SmallGroup(s,n). Then SavePcNormalizedUnitGroup(G) creates the file with the name of the form us&uscore;n.g in the directory unitlib/userdata, and returns true if this file was successfully generated. This file contains the description of the normalized unit group V(KG) of the group algebra of the group G over the field of p elements.

If the order of G is greater than 243, after this you can construct the group V(KG) using 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 unitlib/userdata directory. If there will be no such file there, you will get an error message, otherwise the computation will begin.

If the order of G is less or equal to 243, then the file will be created in the unitlib/userdata directory, but &UnitLib; will continue to use the file with the same name from the appropriate directory in unitlib/data. You can compare these two files to make it sure that they are the same.

WARNINGS:

1. It is important to apply this function to the underlying group G and not to the normalized unit group V(KG).

2. The user should use as an argument only groups from the Small Groups Library of the &GAP; system, constructed with the help of SmallGroup(s,n), otherwise the consistency of data may be lost.
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>
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Service tools
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.
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.