Currently the only subgroups of a group of Lie type that can be constructed are subsystem subgroups.
The subsystem subgroup of the group of Lie type G generated by the standard maximal torus and the root subgroups with roots αa1, ..., αak where a={a1, ..., ak} is a set of integers.
The subsystem subgroup of the group of Lie type G generated by the standard maximal torus and the root subgroups with roots αs1, ..., αsk where s=[s1, ..., sk] 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.
> 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
The direct product of the groups G1 and G2. The two groups must have the same base ring.
The dual of the group of Lie type G, obtained by swapping the roots and coroots.
The soluble radical of the group of Lie type G.
The standard maximal torus of the group of Lie type G.
> 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