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.
> 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)
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.
The commutator g - 1h - 1gh of g and h, where g and h are elements of a group of Lie type.
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].
> 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 an 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 that 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 q powers in the field.[Next][Prev] [Right] [Left] [Up] [Index] [Root]