In this chapter we describe the approach used to store the normalized unit group of the group algebra in the library, and to reconstruct the group \(V(KG)\) from the stored information.
To compute the pc-presentation of the normalized unit group of the modular group algebra of a finite \(p\)-group we used the function PcNormalizedUnitGroup from the LAGUNA package. It uses the algorithm descibed in [Bov98]. See the LAGUNA manual [BKRS] for more details.
When this group is computed, the main idea is to use GAP function CodePcGroup that returns the code for the polycyclic generating sequence of the group, and then to create the group from this code using the GAP function PcGroupCode.
The resulting code could be very long, and to compress it we used the GAP function HexStringInt than returns a string that represents the code with hexa-decimal digits (using A-F as digits 10-15). The inverse translation then can be performed with the GAP function IntHexString. This allowed to save almost 20 MB of space on groups of order 128 and reduce the total size of their database to 90 MB.
For groups of order 128 we decided to compress the library with the gzip program, and then uncompress each file "on fly" when it is requested. This allowed us to reduce the size of their part in the library from 90 to 12 MB, which is already quite reasonable. Of course, there is some little overhead arising from the uncompression and subsequent translation from hexa-decimal notation, but it is neglectible comparatively with the total time of the computation of \(V(KG)\) from scratch.
There is one more thing that needs to be stored together with this code to make it sure that we will correctly identify the underlying group \(G\) of the group algebra \(KG\) with its image in the pc-presentation of the normalized unit group \(V(KG)\).
The group \(G\) is extracted from the GAP Small Groups Library, so it is always the same, unless its description in the library will be changed (and it will be an important task of UnitLib maintaner to update the package in this case!), and here we are safe from inconsistencies.
But the next stage is the computation of generators of the normalized unit group \(V(KG)\), and the first step is the dimension basis of the group \(G\), that can be computed using the LAGUNA function DimensionBasis. To avoid the influence of possible changes in GAP or usage of random methods, we store (in compacted form) the information about the dimension basis of \(G\) in the UnitLib.
All further procedures are implemented inside the LAGUNA package, and their result is uniquely determined and predictable.
For the reader interested in more details, the package contains the file unitlib/lib/genlib.g with the function CreatePcNormalizedUnitGroupsLibrary, that creates library files for groups of a given prime power order, and the file unitlib/lib/genpar.g with the function ParCreatePcNormalizedUnitGroupsLibrary, which is the parallel version of the previous function and must be used with the ParGAP package [Coo].
To reconstruct the normalized unit group \(V(KG)\) from the library, we need only to know the catalogue number of the underlying group \(G\) in the GAP Small Groups Library.
We use the same numbering as in the GAP Small Group Library, so UnitLib finds the appropriate library file(s) and reads from it the code for the polycyclic generating sequence of \(V(KG)\) and the information about the dimension basis of \(G\) used for the computation of this code.
Then \(V(KG)\) is created from the code using the GAP function PcGroupCode (Reference: PcGroupCode). We also create \(G\) using the GAP Small Groups Library.
Now to "glue" the group \(V(KG)\) with the underlying group \(G\) properly, the value of the attribute DimensionBasis (LAGUNA: DimensionBasis) of \(G\) is set in accordance with the data retrieved from the library. This will guarantee the correct construction of NaturalBijectionToPcNormalizedUnitGroup (LAGUNA: NaturalBijectionToPcNormalizedUnitGroup) and NaturalBijectionToNormalizedUnitGroup (LAGUNA: NaturalBijectionToNormalizedUnitGroup) by the LAGUNA package.
It remains now to make only several technical steps, such as constructing the group algebra \(KG\) over the appropriate field \(K\), and storing \(KG\) in the attribute UnderlyingGroupRing (LAGUNA: UnderlyingGroupRing) of \(V(KG)\) and \(V(KG)\) in the attribute PcNormalizedUnitGroup (LAGUNA: PcNormalizedUnitGroup) of \(KG\).
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