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# 5 Additional functionality for matrix groups

### Sections

1. Basic attributes for matrix groups
2. Irreducibility and maximality of matrix groups
3. Primitivity of matrix groups
4. Conjugating matrix groups into smaller fields

This chapter explains some attributes, properties and operations which may be useful for working with matrix groups. Some of these are part of the GAP library and have been included for the sake of completeness, and some are provided by the package IRREDSOL. Note that groups constructed by functions in IRREDSOL already have the appropriate properties and attributes.

## 5.1 Basic attributes for matrix groups

- `DegreeOfMatrixGroup(`G`) F`
- `Degree(`G`) O`
- `DimensionOfMatrixGroup(`G`) A`
- `Dimension(`G`) O`

  This is the degree of the matrix group or, equivalently, the dimension of the natural underlying vector space. See also DimensionOfMatrixGroup.

- `FieldOfMatrixGroup(`G`) A`

  This is the field generated by the matrix entries of the elements of G. See also FieldOfmatrixGroup.

- `CharacteristicOfField(`G`) A`
- `Characteristic(`G`) O`

  This is the characteristic of `FieldOfMatrixGroup` (see FieldOfMatrixGroup).

- `RepresentationIsomorphism(`G`) A`

  This attribute stores an isomorphism H toG, where H is a group in which computations can be carried out more efficiently than in G, and the isomorphism can be evalueated easily. It is usually advantageous to carry out computations in H, and to translate them to G via `RepresentationIsomorphism`(G). (Note that the inverse mapping of `RepresentationIsomorphism` is not required to be efficient.)

  ## 5.2 Irreducibility and maximality of matrix groups

- `IsIrreducibleMatrixGroup(`G`) A`
- `IsIrreducibleMatrixGroup(`G`,` F`) O`
- `IsIrreducible(`G`[,` F`]) O`

  The matrix group G of degree d is irreducible over the field F if no subspace of Fd is invariant under the action of G. If F is not specified, F = `FieldOfMatrixGroup`(G) is assumed.

- `IsAbsolutelyIrreducibleMatrixGroup(`G`) A`

  If present, this attribute is true if G is absolutely irreducible, i. e., irreducible over any extension field of `FieldOfMatrixGroup`(G).

- `IsMaximalAbsolutelyIrreducibleSolvableMatrixGroup(`G`) A`

  This attribute, if present, is true if, and only if, G is absolutely irreducible and maximal among the soluble subgroups of GL(d, F), where d = `DegreeOfMatrixGroup`(G) and F = `FieldOfMatrixGroup`(G).

  ## 5.3 Primitivity of matrix groups

- `MinimalBlockDimensionOfMatrixGroup(`G`) A`
- `MinimalBlockDimensionOfMatrixGroup(`G`,` F`) O`
- `MinimalBlockDimension(`G`[,` F`]) O`

  Let G be a matrix group of degree d over the field F. A decomposition V1 opluscdotsoplusVk of Fd into F-subspaces Vi is a block system of G if the Vi are permuted by the natural action of G. Obviously, all Vi have the same dimension; this is the dimension of the block system V1 opluscdotsoplusVk. The function `MinimalBlockDimensionOfMatrixGroup` returns the minimum of the dimensions of all block systems of G. If F is not specified, F = `FieldOfMatrixGroup`(G) is assumed.

- `IsPrimitiveMatrixGroup(`G`) A`
- `IsPrimitiveMatrixGroup(`G`,` F`) O`
- `IsLinearlyPrimitive(`G`[,` F`]) F`
- `IsPrimitive(`G`[,` F`]) O`

  The matrix group G of degree d is primitive over the field F if it only has the trivial block system Fd or, equivalently, if `MinimalBlockDimensionOfMatrixGroup`(G, F) = d. If F is not specified, F = `FieldOfMatrixGroup`(G) is assumed.

  ## 5.4 Conjugating matrix groups into smaller fields

- `TraceField(`G`) A`

  This is the field generated by the traces of the elements of G. If G is a matrix group over a finite field, then By Brauer's theorem, G has a conjugate which is a matrix group over `TraceField`(G).

- `ConjugatingMatTraceField(`G`) A`

  If bound, this is a matrix x over `FieldOfMatrixGroup`(G) such that G<x> is a matrix group over `TraceField`(G). Currently, there are only methods available for absolutely irreducible groups G (described in GH) and certain trivial cases.

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