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# 6 Advanced information

### Sections

1. Design of the group library
2. Loading and unloading group data manually
3. Loading and unloading recognition data manually

This chapter gives some additional information about the IRREDSOL library. It describes details about the library design (see Design of the group library) and some advanced functions for managing data in the GAP workspace, which may be useful if you run out of memory. See Loading and unloading group data manually and Loading and unloading recognition data manually.

## 6.1 Design of the group library

To avoid redundancy, the package IRREDSOL does not store lists of irreducible subgroups of GL(n, q) but only has lists cal An,q of subgroups of GL(n, q) such that

--
  each group in calAn,q is absolutely irreducible and solvable

--
  calAn,q contains a conjugate of each absolutely irreducible solvable subgroup of

--
  no two groups in calAn,q are conjugate

--
  no conjugate of G is a subgroup of GL(n, q0), where GF(q0) is a proper subfield of GF(q).

briefly say that calAn,q contains, up to conjugacy, all absolutely irreducible solvable subgroups of GL(n,q) which cannot be written over a proper subfield.

These lists are sufficient to reconstruct lists of irreducible soluble subgroups of GL(n, q), since any such subgroup can be obtained from an absolutely irreducible subgroup of GL(n/d, qd), where d divides n, by regarding the underlying Fq^d-vector space as an Fq-vector space. Note that two subgroups of GL(n, q) constructed in that way are conjugate if, and only if, the images of the corresponding subgroups of under Galois automorphisms of Fq^d/Fq are conjugate in GL(n/d, qd). This information also forms part of the IRREDSOL library.

Note that by the Deuring-Noether theorem, two subgroups of GL(n, q) are conjugate in GL(n, q) if, and only if, they are conjugate in GL(n, qd) for some d > 1. This ensures that lists of irreducible subgroups obtained from the calAn,q do not contain conjugate subgroups.

## 6.2 Loading and unloading group data manually

The functions in this section are not required for normal use of the data library. They are only relevant if timing or conservation of memory is an issue.

- `LoadAbsolutelyIrreducibleSolvableGroupData(`n`,` q`) F`

  This function loads the data for GL(n, q) into memory and does some pre-processing. If the data is already loaded, the function does nothing. This function is called automatically when you access the IRREDSOL library, so most users will not need this function.

- `LoadedAbsolutelyIrreducibleSolvableGroupData() F`

  This function returns a list. Each entry consists of an integer n and a set l. The set l contains all prime powers q such that the group data for GL(n, q) is currently in memory.

- `UnloadAbsolutelydIrreducibleSolvableGroupData([n [,q]]) F`

  This function can be used to delete data for irreducible groups from the GAP workspace. If no argument is given, all data will be deleted. If only n is given, all data for degree n (and any q) will be deleted. If n and q are given, only the data for GL(n, q) will be deleted from the GAP workspace. Use this function if you run out of GAP workspace. The data is automatically re-loaded when required.

  ## 6.3 Loading and unloading recognition data manually

  The functions in this section are not required for normal use of the data library. They are only relevant if timing or conservation of memory is an issue.

- `LoadedAbsolutelyIrreducibleSolvableGroupFingerprints() F`

  This function returns a list. Each entry consists of an integer n and a set l. The set l contains all prime powers q such that the recognition data for GL(n, q) is currently in memory.

- `UnloadAbsolutelyIrreducibleSolvableGroupFingerprints([n [,q]]) F`

  This function can be used to delete recognition for irreducible groups from the GAP workspace. If no argument is given, all data will be deleted. If only n is given, all data for degree n (and any q) will be deleted. If n and q are given, only the data for GL(n, q) will be deleted from the GAP workspace. Use this function if you run out of GAP workspace. The data is automatically re-loaded when required.

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