# Building Groups of Lie Type

Currently the only subgroups of a group of Lie type that can be constructed are subsystem subgroups.

## `SubsystemSubgroup(G, a): GrpLie, SetEnum -> RootDtm`

The subsystem subgroup of the group of Lie type $G$ generated by the standard maximal torus and the root subgroups with roots $\alpha_{a_1},\dots,\alpha_{a_k}$ where $a=\{a_1,\dots,a_k\}$ is a set of integers.

## `SubsystemSubgroup(G, s): GrpLie, SeqEnum -> RootDtm`

The subsystem subgroup of the group of Lie type $G$ generated by the standard maximal torus and the root subgroups with roots $\alpha_{s_1},\dots,\alpha_{s_k}$ where $s=[s_1,\dots,s_k]$ is a *sequence* of integers. In this version the roots must be simple in the root subdatum (i.e. none of them may be a summand of another) otherwise an error is signalled. The simple roots will appear in the subdatum in the given order.

## `Example: Root Subdata (ex-0494af)`

```magma
> G := GroupOfLieType("A4",Rationals());
> PositiveRoots(G);
{@
    (1 0 0 0),
    (0 1 0 0),
    (0 0 1 0),
    (0 0 0 1),
    (1 1 0 0),
    (0 1 1 0),
    (0 0 1 1),
    (1 1 1 0),
    (0 1 1 1),
    (1 1 1 1)
@}
> H := SubsystemSubgroup(G, [6,1,4]);
> H;
H: Group of Lie type A3 over Rational Field
> PositiveRoots(H);
{@
    (0 1 1 0),
    (1 0 0 0),
    (0 0 0 1),
    (1 1 1 0),
    (0 1 1 1),
    (1 1 1 1)
@}
> h := elt<H|<2,2>,1>;
> h;  G!h;
x2(2) n1
x1(2) ( 1 -1  1 -1) n2 n3 n2

```

## `DirectProduct(G1, G2): GrpLie, GrpLie -> GrpLie`

The direct product of the groups $G_1$ and $G_2$. The two groups must have the same base ring.

## `Dual(G): GrpLie -> GrpLie`

The dual of the group of Lie type $G$, obtained by swapping the roots and coroots.

## `SolubleRadical(G): GrpLie -> GrpLie`

The soluble radical of the group of Lie type $G$.

## `StandardMaximalTorus(G): GrpLie -> GrpLie`

The standard maximal torus of the group of Lie type $G$.

## `Example: Direct Product Dual Radical (ex-c4d20b)`

```magma
> G1 := GroupOfLieType( "A5", GF(7) );
> G2 := GroupOfLieType( "B4", GF(7) );
> DirectProduct(G1, Dual(G2));
$: Group of Lie type A5 C4 over Finite field of size 7
>
> G := GroupOfLieType(StandardRootDatum("A",3), GF(17));
> SolubleRadical(G);
$: Torus group of Dimension 1 over Finite field of size 17

```