The product of two elements of a group of Lie type.
If the Normalising flag is set for the group, then the product is normalised using the algorithms of [CMT04], [CHM08].
Otherwise, the words are just concatenated.
If the
Normalising flag is set, the product is normalised,
otherwise multiplication is just concatenation.
> G := GroupOfLieType("G2", GF(3) : Normalising:=false );
> V := VectorSpace(GF(3),2);
> g := elt< G | 1,2,1,2, V![2,2], <1,2>,<5,1> >;
> h := elt< G | <3,2>, V![1,2], 1 >;
> g*h;
n1 n2 n1 n2 (2 2) x1(2) x5(1) x3(2) (1 2) n1
> H := GroupOfLieType("G2", GF(3) : Normalising:=true );
> g := elt< H | 1,2,1,2, V![2,2], <1,2>,<5,1> >;
> h := elt< H | <3,2>, V![1,2], 1 >;
> g*h;
x2(1) x3(1) (1 2) n1 n2 n1 n2 n1 x4(1)
Inverse(G) : GrpLieElt -> GrpLieElt
The inverse of the element g of a group of Lie type.
The nth power of the element g of a group of Lie type.
The conjugate h - 1gh, where g and h are elements of a group of Lie type.
Commutator(g, h) : GrpLieElt, GrpLieElt -> GrpLieElt
The commutator g - 1h - 1gh of g and h,
where g and h are elements of a group of Lie type.
Normalize(simg) : GrpLieElt ->
Normalise(g) : GrpLieElt -> GrpLieElt
Normalize(g) : GrpLieElt -> GrpLieElt
Normalise the element g of a group of Lie type G. The procedural form is slightly more efficient
than the functional form. If the Normalise flag is set for G,
this operation has no effect.
This uses the algorithms of [CMT04], [CHM08].
Arithmetic in groups of Lie type.
> k<z> := GF(4);
> G := GroupOfLieType("C3", k);
> V := VectorSpace(k, 3);
> g := elt< G | 1,2,3, <3,z>,<4,z^2>, V![1,z^2,1] >;
> g;
n1 n2 n3 x3(z) x4(z^2) ( 1 z^2 1)
> h := elt< G | [0,1,z,1,0,z^2,1,1,z] >;
> h;
x2(1) x3(z) x4(1) x6(z^2) x7(1) x8(1) x9(z)
> g * h^-1;
x3(1) x5(z) x6(z^2) x8(1) (z^2 z^2 z) n1 n2 n3 x3(z^2) x5(z^2)
> g^3;
x3(z) x5(1) x7(z^2) x8(z^2) ( 1 1 z) n1 n2 n3 n1 n2 n3 n1 n2 n3 x1(1)
x2(z^2) x3(1) x4(z) x7(z) x9(z)
Given an element g of a group of Lie type the Bruhat decomposition
of g is returned. The function returns elements u,
h, /dot w, u' with the properties described in
Subsection Twisted Groups of Lie type and so that g=uh/dot wu'.
> k<z> := GF(4);
> G := GroupOfLieType("C3", k);
> V := VectorSpace(k, 3);
> g := elt< G | 1,2,3, <3,z>,<4,z^2>, V![1,z^2,1] >;
> Normalise(g);
x7(z^2) x8(z^2) (z^2 z^2 z) n1 n2 n3 x3(1) x6(z)
> u, h, w, up := Bruhat(g);
> u; h; w; up;
x7(z^2) x8(z^2)
(z^2 z^2 z)
n1 n2 n3
x3(1) x6(z)
The multiplicative Jordan decomposition of the element x of the group of Lie type.
Given a semisimple element g in a finite group of Lie type, return a torus
element t and conjugator x such that t=xgx - 1. The elements returned
may be defined over a larger field than the input element.
Given a semisimple element g in a finite group of Lie type, return a
Borel element b
and conjugator x such that b=xgx - 1.
The elements returned may be defined over a larger field that the input element.
Although any element of a group of Lie type can be conjugated into the Borel
subgroup, this function is currently only implemented for semisimple elements.
Given an element c in a finite group of Lie type and q a power of the
characteristic, return a solution a of the Lang equation c = a - F a.
Here F is the Frobenius automorphism gotten by taking qth powers in
the field.
V2.28, 13 July 2023