Elements of PSL₂(R)

Creation

G ! x : GrpPSL2, . -> GrpPSL2

G ! x : GrpPSL2, RngIntElt -> GrpPSL2

G ! x : GrpPSL2, GrpMatElt -> GrpPSL2

G ! x : GrpPSL2, AlgMatElt -> GrpPSL2

G ! x : GrpPSL2, GrpPSL2Elt -> GrpPSL2

If x is a sequence x = [a, b, c, d] of elements in the base ring of G, this function returns pmatrix(a & b
c & d), provided this is an element of G. If x is an integer the identity matrix is returned. If x is a matrix, it is coerced into G if possible.

Random(G,m) : GrpPSL2, RngIntElt -> GrpPSL2Elt

Returns a random element of the projective linear group G, with m determining the size of the coefficients.

Membership and Equality Testing

g eq h : GrpPSL2Elt, GrpPSL2Elt -> BoolElt

For g and h elements of PSL₂(Z), returns true if g, h have compatible coefficient rings and if g = h, false otherwise. Since the group is projective, returns true if the matrices are equal up to a nonzero scalar multiple.

IsEquivalent(g,h,G) : GrpPSL2Elt, GrpPSL2Elt, GrpPSL2 -> BoolElt

For g and h elements of PSL₂(Z), returns true if g and h are defined of the same field, and if Gg = Gh, i.e. if gh ^{- 1}∈G.

g in G : GrpPSL2Elt, GrpPSL2 -> BoolElt

For g an elements of PSL₂(Z), returns true if g is in the congruence subgroup G, false otherwise.

Basic Functions

For a matrix g in a congruence subgroup, and an integer n, returns gⁿ.

Eltseq(g) : GrpPSL2Elt -> SeqEnum

Returns the sequence of four numbers which are the entries of the matrix g.

g * h : GrpPSL2Elt, GrpPSL2Elt -> GrpPSL2Elt

If g and h have the same parent then this returns their product.

g ^ n : GrpPSL2Elt, RngIntElt -> GrpPSL2Elt

For a matrix g and integer n returns gⁿ.

> // examples of defining matrix elements of congruence subgroups:
> 
> G := PSL2(Integers());
> G![2,0,0,2];
[1 0]
[0 1]

> H := CongruenceSubgroup([2,3,6]);
> H![7,6,8,7];                   
[7 6]
[8 7]