# Reductive Groups

- [Introduction](introduction.md)

  - [Definitions and Background](introduction.md#definitions-and-background)

  - [Categories](introduction.md#categories)

- [Creation of Reductive Groups](creation.md)

  - [`ReductiveGroup(G0, Comp): GrpLie, Grp → GrpRed`](creation.md#function-reductivegroup-grplie-grp)

  - [`ReductiveGroup(group_data): List → GrpRed`](creation.md#function-reductivegroup-list)

  - [`SymplecticGroup(V): SpcPlr → GrpRed`](creation.md#function-symplecticgroup-spcplr)

  - [`SymplecticGroup(n, F): RngIntElt, Fld → GrpRed`](creation.md#function-symplecticgroup-rngintelt-fld)

  - [`SymplecticGroup(Q): AlgMatElt[Fld] → GrpRed`](creation.md#function-symplecticgroup-algmatelt-fld)

  - [`SymplecticGroup(Q): AlgMatElt[Rng] → GrpRed`](creation.md#function-symplecticgroup-algmatelt-rng)

  - [`OrthogonalGroup(V): SpcPlr → GrpRed`](creation.md#function-orthogonalgroup-spcplr)

  - [`OrthogonalGroup(n, F): RngIntElt, Fld → GrpRed`](creation.md#function-orthogonalgroup-rngintelt-fld)

  - [`OrthogonalGroup(Q): AlgMatElt[Fld] → GrpRed`](creation.md#function-orthogonalgroup-algmatelt-fld)

  - [`OrthogonalGroup(Q): AlgMatElt[Rng] → GrpRed`](creation.md#function-orthogonalgroup-algmatelt-rng)

  - [`SpecialOrthogonalGroup(V): SpcPlr → GrpRed`](creation.md#function-specialorthogonalgroup-spcplr)

  - [`SpecialOrthogonalGroup(n, F): RngIntElt, Fld → GrpRed`](creation.md#function-specialorthogonalgroup-rngintelt-fld)

  - [`SpecialOrthogonalGroup(Q): AlgMatElt[Fld] → GrpRed`](creation.md#function-specialorthogonalgroup-algmatelt-fld)

  - [`SpecialOrthogonalGroup(Q): AlgMatElt[Rng] → GrpRed`](creation.md#function-specialorthogonalgroup-algmatelt-rng)

  - [`UnitaryGroup(V): SpcPlr → GrpRed`](creation.md#function-unitarygroup-spcplr)

  - [`UnitaryGroup(n, F): RngIntElt, Fld → GrpRed`](creation.md#function-unitarygroup-rngintelt-fld)

  - [`UnitaryGroup(Q): AlgMatElt[Fld] → GrpRed`](creation.md#function-unitarygroup-algmatelt-fld)

  - [`UnitaryGroup(Q): AlgMatElt[Rng] → GrpRed`](creation.md#function-unitarygroup-algmatelt-rng)

- [Properties of Reductive Groups](properties.md)

  - [`ConnectedComponent(G): GrpRed → GrpRed`](properties.md#function-connectedcomponent-grpred)

  - [`ComponentGroup(G): GrpRed → GrpRed`](properties.md#function-componentgroup-grpred)

  - [`SplittingField(G): GrpRed → Fld`](properties.md#function-splittingfield-grpred)

  - [`FieldOfDefinition(G): GrpRed → Fld`](properties.md#function-fieldofdefinition-grpred)

  - [`BaseRing(G): GrpRed → Fld`](properties.md#function-basering-grpred)

  - [`CartanName(G): GrpRed → MonStgElt`](properties.md#function-cartanname-grpred)

  - [`InnerForm(G, i): GrpRed, RngIntElt → SpcPlr`](properties.md#function-innerform-grpred-rngintelt)

  - [`InnerForms(G): GrpRed → [ SpcPlr ]`](properties.md#function-innerforms-grpred)

  - [`Dimension(G): GrpRed → RngIntElt`](properties.md#function-dimension-grpred)

  - [`Degree(G): GrpRed → RngIntElt`](properties.md#function-degree-grpred)

  - [`Rank(G): GrpRed → RngIntElt`](properties.md#function-rank-grpred)

  - [Predicates](properties.md#predicates)

    - [`IsConnected(G): GrpRed → BoolElt`](properties.md#function-isconnected-grpred)

    - [`IsOrthogonal(G): GrpRed → BoolElt`](properties.md#function-isorthogonal-grpred)

    - [`IsSpecialOrthogonal(G): GrpRed → BoolElt`](properties.md#function-isspecialorthogonal-grpred)

    - [`IsSymplectic(G): GrpRed → BoolElt`](properties.md#function-issymplectic-grpred)

    - [`IsUnitary(G): GrpRed → BoolElt`](properties.md#function-isunitary-grpred)

    - [`IsCompact(G): GrpRed → BoolElt`](properties.md#function-iscompact-grpred)

- [Operations on Reductive Groups](ops.md)

  - [`GetSplitPrimeWithSquare(G): GrpRed → RngIntElt`](ops.md#function-getsplitprimewithsquare-grpred)

  - [Base change](ops.md#base-change)

    - [`ChangeRing(G, S): GrpRed, Rng → GrpRed`](ops.md#function-changering-grpred-rng)

- [Elements of Reductive Groups](elements.md)

  - [`Parent(g): GrpRedElt → GrpRed`](elements.md#function-parent-grpredelt)

  - [Arithmetic of Elements](elements.md#arithmetic-of-elements)

    - [`x * y: GrpRedElt, GrpRedElt → GrpRedElt`](elements.md#operation-op-times-grpredelt-grpredelt)

    - [`x ^ n: GrpRedElt, RngIntElt → GrpRedElt`](elements.md#operation-op-pow-grpredelt-rngintelt)

  - [Other Operations with Elements](elements.md#other-operations-with-elements)

    - [`ElementToSequence(x): GrpRedElt → SeqEnum`](elements.md#function-elementtosequence-grpredelt)

    - [`Eltseq(x): GrpRedElt → SeqEnum`](elements.md#function-eltseq-grpredelt)