Finite imprimitive linear groups of prime degree

In an earlier paper the authors have classified the nonsolvable primitive linear groups of prime degree over C. 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. Mathematics Subject Classification 2000: 20H20