Quasiprimitive Linear Groups with Quadratic Elements

A semisimple complex linear transformation with exactly two distinct eigenvalues is called a quadratic element. In this paper the finite irreducible linear groups generated by quadratic elements of order 3 for which the multiplicities of the eigenvalues are distinct are determined. This is used to obtain bounds on the degree of an irreducible primitive linear group containing such elements. There are no irreducible primitive linear groups containing quadratic elements of order 4 with eigenvalues {1, i} with distinct multiplicities where i is a primitive fourth root of 1.