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# 2 Accessing the data library

### Sections

This chapter describes the design of the IRREDSOL group library (see Section Design of the group library) and the various ways of accessing groups in the data library. It is possible to construct individual groups in the group library (see Section Low level access functions), or to search for groups with certain properties (see Section Finding matrix groups with given properties). Finally, there are functions for loading and unloading group data manually (see Section Loading and unloading group data manually).

## 2.1 Design of the group library

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

bullet
each group in calAn,q is absolutely irreducible and soluble

bullet
calAn,q contains a conjugate of each absolutely irreducible soluble subgroup of GL(n, q)

bullet
no two groups in calAn,q are conjugate in GL(n, q)

bullet
each group in calAn,q has trace field Fq.

(For n = 1, such lists are not actually stored but are computed when required.)

We will briefly say that calAn,q contains, up to conjugacy, all absolutely irreducible soluble subgroups of GL(n,q) with trace field Fq. Here, the trace fieldof a subgroup G of GL(n, q) is the field generated by the traces of the elements of G. By a theorem of Brauer, an irreducible subgroup of GL(n, q) with trace field Fq_0 has a conjugate lying in GL(n, q0). See also `TraceField` (TraceField) and `ConjugatingMatTraceField` (ConjugatingMatTraceField).

Note that by the Deuring-Noether theorem, two subgroups of GL(n, q0) are conjugate in GL(n, q0) if, and only if, they are conjugate in GL(n, q). Therefore, we obtain, up to conjugacy, all absolutely irreducible soluble subgroups of GL(n, q) by forming the union of the calAn, q_0, where Fq_0 runs over all subfields of Fq.

The lists calAn, q are also sufficient to reconstruct lists of all irreducible soluble subgroups of GL(n, q). Let d be a divisor of n, and let G be an absolutely irreducible subgroup of GL(n/d, qd). By regarding the underlying Fq^d-vector space of G as an Fq-vector space, we obtain an irreducible subgroup G^* of GL(n, q) with splitting field Fq^d. Up to conjugacy, all irreducible subgroups of GL(n, q) arise in that way. If two subgroups G1 and G2 of GL(n, q) are constructed from subgroups G1^* and G2^* of GL(n/d, qd), then G1 and G2 are conjugate if, and only if, there exists a Galois automophism sigma of Fq^d/Fq such that (G1^*)^sigma and G2^* are conjugate in GL(n/d, qd). See, e. g., DH, Theorem B 5.15. The latter information has been precomputed and is also part of the IRREDSOL library.

Note that all of the arguments above apply to nonsoluble groups as well.

## 2.2 Low level access functions

The access functions described in this section allow to check for the availability of data and to construct irreducible groups in the IRREDSOL group library.

• `IsAvailableIrreducibleSolubleGroupData(`n`, `q`) F`
• `IsAvailableIrreducibleSolvableGroupData(`n`, `q`) F`

This function tests whether the irreducible soluble subgroups of GL(n,q) with trace field Fq are part of the IRREDSOL library.

• `IndicesIrreducibleSolubleMatrixGroups(`n`, `q`, `d`) F`
• `IndicesIrreducibleSolvableMatrixGroups(`n`, `q`, `d`) F`

Let n and d be positive integers and q a prime power. This function returns a set of integers parametrising the groups in the IRREDSOL library which are subgroups of GL(n,q) with trace field Fq and splitting field Fq^d. This set is empty unless d divides n. An error is raised if the relevant data is not available, cf. IsAvailableIrreducibleSolubleGroupData.

```gap> IndicesIrreducibleSolubleMatrixGroups(6, 2, 2);
[ 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12 ]
```

• `IrreducibleSolubleMatrixGroup(`n`, `q`, `d`, `k`) F`
• `IrreducibleSolvableMatrixGroup(`n`, `q`, `d`, `k`) F`

Let n, d and k be positive integers and q a prime power. This function returns the irreducible soluble subgroup of GL(n,q) with trace field Fq, splitting field Fq^d, and index k. An error is raised if the relevant data is not available, or if k is not in `IndicesIrreducibleSolubleMatrixGroups`(n, q, d); cf. IsAvailableIrreducibleSolubleGroupData and IndicesIrreducibleSolubleMatrixGroups. The groups returned have the attributes and properties described in Chapter Additional functionality for matrix groups set to their appropriate values.

• `IsAvailableAbsolutelyIrreducibleSolubleGroupData(`n`, `q`) F`
• `IsAvailableAbsolutelyIrreducibleolvableGroupData(`n`, `q`) F`

This function tests whether the absolutely irreducible soluble subgroups of GL(n,q) with trace field Fq are in the IRREDSOL library.

• `IndicesMaximalAbsolutelyIrreducibleSolubleMatrixGroups(`n`, `q`) F`
• `IndicesMaximalAbsolutelyIrreducibleSolvableMatrixGroups(`n`, `q`) F`

Let n be a positive integer and q a prime power. This function returns a set of integers parametrising those subgroups of GL(n, q) in the IRREDSOL library that are maximal among the absolutely irreducible soluble subgroups with trace field Fq and that are maximal with respect to being soluble. An error is raised if the relevant data is not available (see IsAvailableAbsolutelyIrreducibleSolubleGroupData for information how to check this first). An integer k in the list return corresponds to `IrreducibleSolubleMatrixGroup`(n, q, 1, k), which is the same group as `AbsolutelyIrreducibleSolubleMatrixGroup`(n, q, k).

```gap> inds := IndicesMaximalAbsolutelyIrreducibleSolubleMatrixGroups(2,3);
[ 2 ]
gap> IrreducibleSolubleMatrixGroup(2,3,1,2) = GL(2,3); # it is the whole GL
true
```

• `IndicesAbsolutelyIrreducibleSolubleMatrixGroups(`n`, `q`) F`
• `AbsolutelyIrreducibleSolubleMatrixGroup(`n`, `q`, `k`) F`

These functions are deprecated. Please use `IndicesIrreducibleSolubleMatrixGroups`(n, q, 1) and `IrreducibleSolubleMatrixGroup`(n, q, 1, k) instead.

## 2.3 Finding matrix groups with given properties

This section describes three functions (`AllIrreducibleSolubleMatrixGroups`, `OneIrreducibleSolubleMatrixGroup`, `IteratorIrreducibleSolubleMatrixGroups`) which allow to find matrix groups with prescribed properties. Using these functions can be more efficient than to construct each group in the library using the functions in Section Low level access functions because they can access additional information about a group in the IRREDSOL library before actually constructing the group. See the discussion following the description of `AllIrreducibleSolubleMatrixGroups` for details.

• `AllIrreducibleSolubleMatrixGroups(`func_1`, `arg_1`, `func_2`, `arg_2`, ...) F`
• `AllIrreducibleSolvableMatrixGroups(`func_1`, `arg_1`, `func_2`, `arg_2`, ...) F`
• `AllIrredSolMatGroups(`func_1`, `arg_1`, `func_2`, `arg_2`, ...) F`

This function returns a list of all irreducible soluble matrix groups G in the IRREDSOL library for which the return value of funci(G) lies in arg_i. The arguments func_1, func_2, ..., must be GAP functions which take a matrix group as their only argument and return a value, and arg_1, arg_2, ..., must be lists. If arg_i is not a list, arg_i is replaced by the list `[`arg_i`]`. The functions `DegreeOfMatrixGroup` and `FieldOfMatrixGroup` (or their equivalents, see below) must be among the func_i. `TraceField`). Note that all groups in the data library have the property that `TraceField`(G) = `FieldOfMatrixGroup`(G); see Section Design of the group library for details.

Note that there is also a function `IteratorIrreducibleSolubleMatrixGroups` (see IteratorIrreducibleSolubleMatrixGroups) which allows to run through the list produced by `AllIrreducibleSolubleMatrixGroups` without having to store all of the groups simultaneously.

The following functions func_i are handled particularly efficiently, because the return values of these functions can be read off the IRREDSOL library without actually constructing the relevant matrix group. For the definitions of these functions, see Chapter Additional functionality for matrix groups.

bullet
`DegreeOfMatrixGroup` (or `Degree`, `Dimension`, `DimensionOfMatrixGroup`),
bullet
`CharacteristicOfField` (or `Characteristic`)
bullet
`FieldOfMatrixGroup` (or `Field` or `TraceField`)
bullet
`Order` (or `Size`)
bullet
`IsMaximalAbsolutelyIrreducibleSolubleMatrixGroup`
bullet
`IsAbsolutelyIrreducibleMatrixGroup` (or `IsAbsolutelyIrreducible`)
bullet
`MinimalBlockDimensionOfMatrixGroup` (or `MinimalBlockDimension`)
bullet
`IsPrimitiveMatrixGroup` (or `IsPrimitive`, `IsLinearlyPrimitive`)

The groups G passed to the func_i and the groups returned have the attributes and properties described in Chapter Additional functionality for matrix groups set to their appropriate values. Note that you may speed up computations in G by using an isomorphic copy of G, which can be obtained via `RepresentationIsomorphism` (see RepresentationIsomorphism).

```# get just those groups with trace field GF(9)
gap> l := AllIrreducibleSolubleMatrixGroups(Degree, 1, Field, GF(9));;
gap> List(l, Order);
[ 4, 8 ]

# get all irreducible subgroups
gap> l := AllIrreducibleSolubleMatrixGroups(Degree, 1, Field, Subfields(GF(9)));;
gap> List(l, Order);
[ 1, 2, 4, 8 ]

# get only maximal absolutely irreducible ones
gap> l := AllIrreducibleSolubleMatrixGroups(Degree, 4, Field, GF(3),
>             IsMaximalAbsolutelyIrreducibleSolubleMatrixGroup, true);;
gap> SortedList(List(l, Order));
[ 320, 640, 2304, 4608 ]

# get only absolutely irreducible groups
gap> l := AllIrreducibleSolubleMatrixGroups(Degree, 4, Field, GF(3),
> IsAbsolutelyIrreducibleMatrixGroup, true);;
gap> Collected(List(l, Order));
[ [ 20, 1 ], [ 32, 7 ], [ 40, 2 ], [ 64, 10 ], [ 80, 2 ], [ 96, 6 ],
[ 128, 9 ], [ 160, 3 ], [ 192, 9 ], [ 256, 6 ], [ 288, 1 ], [ 320, 2 ],
[ 384, 4 ], [ 512, 1 ], [ 576, 3 ], [ 640, 1 ], [ 768, 1 ], [ 1152, 4 ],
[ 2304, 3 ], [ 4608, 1 ] ]
```

• `OneIrreducibleSolubleMatrixGroup(`func_1`, `arg_1`, `func_2`, `arg_2`, ...) F`
• `OneIrreducibleSolvableMatrixGroup(`func_1`, `arg_1`, `func_2`, `arg_2`, ...) F`
• `OneIrredSolMatGroup(`func_1`, `arg_1`, `func_2`, `arg_2`, ...) F`

This function returns a matrix group G from the IRREDSOL library such that funci(G) lies in arg_i, or `fail` if no such group exists. The arguments func_1, func_2, ..., must be GAP functions taking one argument and returning a value, and arg_1, arg_2, ..., must be lists. If arg_i is not a list, arg_i is replaced by the list `[`arg_i`]`. The functions `DegreeOfMatrixGroup` and `FieldOfMatrixGroup` (or their equivalents, see below) must be among the func_i. Note that all groups in the data library have the property that `TraceField`(G) = `FieldOfMatrixGroup`(G); see Section Design of the group library for details. The groups passed to the func_i and the groups returned have the attributes and properties described in Chapter Additional functionality for matrix groups set to their appropriate values.

• `IteratorIrreducibleSolubleMatrixGroups(`func_1`, `arg_1`, `func_2`, `arg_2`, ...) F`
• `IteratorIrreducibleSolvableMatrixGroups(`func_1`, `arg_1`, `func_2`, `arg_2`, ...) F`
• `IteratorIrredSolMatGroups(`func_1`, `arg_1`, `func_2`, `arg_2`, ...) F`

This function returns an iterator which runs through the list of all matrix groups G in the IRREDSOL library such that funci(G) lies in arg_i. The arguments func_1, func_2, ..., must be GAP functions taking one argument and returning a value, and arg_1, arg_2, ..., must be lists. If arg_i is not a list, arg_i is replaced by the list `[`arg_i`]`. The functions `DegreeOfMatrixGroup` and `FieldOfMatrixGroup` (or their equivalents, see below) must be among the func_i.

Using

`IteratorIrreducibleSolubleMatrixGroups`(func_1, arg_1, func_2, arg_2, ...))

is functionally equivalent to

`Iterator`(`AllIrreducibleSolubleMatrixGroups`(func_1, arg_1, func_2, arg_2, ...))

(see Section Iterators in the GAP reference manual for details) but, unlike `AllIrreducibleSolubleMatrixGroups`, does not store all relevant matrix groups at the same time. This may save a considerable amount of memory.

The data required by the IRREDSOL library is loaded into GAP's workspace automatically whenever required, but is never unloaded automatically. The functions described in this and the following section describe how to load and unload this data manually. They are only relevant if timing or conservation of memory is an issue.

• `LoadAbsolutelyIrreducibleSolubleGroupData(`n`, `q`) F`
• `LoadAbsolutelyIrreducibleSolvableGroupData(`n`, `q`) F`
• `LoadAbsoIrredSolGroupData(`n`, `q`) F`

This function loads the data for GL(n, q) into the GAP workspace 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.

• `LoadedAbsolutelyIrreducibleSolubleGroupData() F`
• `LoadedAbsolutelyIrreducibleSolvableGroupData() F`
• `LoadedAbsoIrredSolGroupData(`n`, `q`) 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.

• `UnloadAbsolutelyIrreducibleSolubleGroupData([`n` [, `q`]]) F`
• `UnloadAbsolutelyIrreducibleSolvableGroupData([`n` [, `q`]]) F`
• `UnloadAbsoIrredSolGroupData(`n`, `q`) F`

This function can be used to delete data for GL(n, q) 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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IRREDSOL manual
February 2017