# Properties of Groups of Lie Type

## `IsFinite(G): GrpLie -> BoolElt`

Return `true` if and only if the group of Lie type $G$ is finite.

## `IsAbelian(G): GrpLie -> BoolElt`

Returns `true` if the group of Lie type $G$ is abelian.

## `IsSimple(G): GrpLie -> BoolElt`

Returns `true` if the group of Lie type $G$ is a simple group as an algebraic group, ie, $G$ has no proper *connected* normal subgroups. This is true if, and only if, the underlying root datum is irreducible. Note that this does not usually mean that $G$ is simple as an abstract group. In previous releases of Magma this function was incorrectly called `IsIrreducible`.

## `IsSimplyLaced(G): GrpLie -> BoolElt`

Returns `true` if the group of Lie type $G$ is simply laced, i.e. its Dynkin diagram contains no multiple bonds.

## `IsSemisimple(G): GrpLie -> BoolElt`

Returns `true` if the group of Lie type $G$ is semisimple.

## `IsAdjoint(G): GrpLie -> BoolElt`

Returns `true` if, and only if, the group of Lie type $G$ is adjoint (i.e. the isogeny group is trivial).

## `IsWeaklyAdjoint(G): GrpLie -> BoolElt`

Returns `true` if, and only if, the group of Lie type $G$ is weakly adjoint, i.e. its isogeny group is isomorphic to ${\mathbb{Z}}^n$, where $n$ is the difference between the rank and the semisimple rank of $G$. Note that if $G$ is semisimple then this function is identical to [`IsAdjoint`](#function-isadjointg).

## `IsSimplyConnected(G): GrpLie -> BoolElt`

Returns `true` if, and only if, the group of Lie type $G$ is simply connected (i.e. the isogeny group is equal to the fundamental group, i.e. the coisogeny group is trivial).

## `IsWeaklySimplyConnected(G): GrpLie -> BoolElt`

Returns `true` if, and only if, the group of Lie type $G$ is weakly simply connected, i.e. its coisogeny group is isomorphic to ${\mathbb{Z}}^n$, where $n$ is the difference between the rank and the semisimple rank of $G$. Note that if $G$ is semisimple then this function is identical to [`IsSimplyConnected`](#function-issimplyconnectedg).

## `IsSplit(G): GrpLie -> BoolElt`

Returns `true` if and only if the group of Lie type $G$ is split.

## `IsTwisted(G): GrpLie -> BoolElt`

Returns `true` if and only if the group of Lie type $G$ is twisted.