Parametrization of all homomorphisms fromPGL(2Z) intoPGL{2,q)

摘要

LetPL(Fq) denote the projective line over a Galois fieldFq. Consider PSL (2, Z ) as a free product of two cyclic groups x and y of orders 2 and 3. We have shown that any homomorphism fromPSL(2,Z) intoPGL(2,q) can be extended to a homomorphism fromPGL(2Z) intoPGL(2q) except in the case where the order of the image ofxyis 6 but the images ofxandydo not commute inPGL(2q). It has been shown also that every element inPGL(2,q), not of order 1,2 , or 6, is the image ofxyunder some non-degenerate homomorphism. We have parametrized the conjugacy classes of non-degenerate homomorphisms #x3B1; with the non-trivial elements ofFq. Due to this parametrization we have developed a useful mechanism by which one can construct.