Finite imprimitive linear groups of prime degree

In an earlier paper the authors classified the nonsolvable primitive linear groups of prime degree over . The present paper deals with the classification of the nonsolvable imprimitive linear groups of prime degree (equivalently, the irreducible monomial groups of prime degree). If G is a monomial group of prime degree r, then there is a projection π of G onto a transitive group H of permutation matrices with a kernel A consisting of diagonal matrices. The transitive permutation groups of prime degree are known, so the classification reduces to (i) determining the possible diagonal groups A for a given group H of permutation matrices; (ii) describing the possible extensions which might occur for given A and H; and (iii) determining when two of these extensions are conjugate in the general linear group. We prove that for given nonsolvable H there is a finite set Φ(r,H) of diagonal groups such that all monomial groups G with π(G)=H can be determined in a simple way from the monomial groups which are extensions of A Φ(r,H) by H, and calculate Φ(r,H) in many cases. We also show how the problem of determining conjugacy in the general case is reduced to solving this problem when A Φ(r,H). In general, the results hold over any algebraically closed field with modifications required in the case of a few small characteristics.