# Properties of Reductive Groups

## `ConnectedComponent(G): GrpRed -> GrpRed`

The connected component of the identity, $G^{\circ}$.

## `ComponentGroup(G): GrpRed -> GrpRed`

The component group of $G$.

## `SplittingField(G): GrpRed -> Fld`

The field over which $G$ splits.

## `FieldOfDefinition(G): GrpRed -> Fld`

## `BaseRing(G): GrpRed -> Fld`

The field of definition of $G$.

## `CartanName(G): GrpRed -> MonStgElt`

A string concatenating the Cartan types of all simple factors and tori of the connected component of $G$.

## `InnerForm(G, i): GrpRed, RngIntElt -> SpcPlr`

The inner form corresponding to the $i$-th simple factor.

## `InnerForms(G): GrpRed -> [ SpcPlr ]`

The inner forms defining the simple factors of $G$.

## `Dimension(G): GrpRed -> RngIntElt`

## `Degree(G): GrpRed -> RngIntElt`

Returns $n$ such that $G$ embeds into ${\rm GL}_n$ through the representation via its inner forms.

## `Rank(G): GrpRed -> RngIntElt`

The rank of $G$.

## Predicates

### `IsConnected(G): GrpRed -> BoolElt`

Returns `true` iff $G$ is connected.

### `IsOrthogonal(G): GrpRed -> BoolElt`

Returns `true` iff $G$ is an orthogonal group.

### `IsSpecialOrthogonal(G): GrpRed -> BoolElt`

Returns `true` iff $G$ is a special orthogonal group.

### `IsSymplectic(G): GrpRed -> BoolElt`

Returns `true` iff $G$ is a symplectic group.

### `IsUnitary(G): GrpRed -> BoolElt`

Returns `true` iff $G$ is a unitary group.

### `IsCompact(G): GrpRed -> BoolElt`

Returns `true` iff $G$ is a compact form. Only relevant for groups defined over number fields.