In fact this set of rational numbers is dense in . The discriminant of equals   and this is not in general a perfect square. Solutions for alternating groups must be handled differently for odd and even degrees. Let  Under this substitution the discriminant of equals  which is a perfect square when is odd. Let:  Under this substitution the discriminant of equals:  which is a perfect square when is even. Again, Hilbert's irreducibility theorem implies the existence of infinitely many specializations whose Galois groups are alternating groups. Suppose that are conjugacy classes of a finite group , and be the set of -tuples of such that is in and the product is trivial. Then is called rigid if it is nonempty, acts transitively on it by conjugation, and each element of generates . This can be used to show that many finite simple groups, including the monster group, are Galois groups of extensions of the rationals. The monster group is generated by a triad of elements of orders , , and . All such triads are conjugate. The prototype for rigidity is the symmetric group , which is generated by an -cycle and a transposition whose product is an -cycle. The construction in the preceding section used these generators to establish a polynomial's Galois group. Let be any integer. A lattice in the complex plane with period ratio has a sublattice with period ratio . The latter lattice is one of a finite set of sublattices permuted by the modular group , which is based on changes of basis for . Let denote the elliptic modular function of Felix Klein. Define the polynomial as the product of the differences over the conjugate sublattices. As a polynomial in , has coefficients that are polynomials over in . On the conjugate lattices, the modular group acts as . It follows that has Galois group isomorphic to over . Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing to polynomials with Galois group over . The groups include infinitely many non-solvable groups.