NILPOTENT PRIMITIVE LINEAR GROUPS OVER FINITE FIELDS A. S. DETINKO AND D. L. FLANNERY 1. Introduction In this paper we investigate the structure of groups as in the title. Our work builds on work of several other authors, namely Konyuh [5], Leedham-Green and Plesken [6], and Zalesskii [10], who have described the abstract isomorphism types of the groups. We obtain more detailed descriptions, in particular explaining how group structure depends on the existence of an abelian primitive subgroup. Additionally we show that isomorphism type of a group completely determines its conjugacy class in the relevant general linear group. A brief outline of the paper now follows. In Section 2 we review standard material on abelian (cyclic) irreducible linear groups. In Section 3 fundamental structural results are given. In Section 4 nilpotent primitive linear groups of degree 2 are classified up to conjugacy, and then groups of degree greater than 2 are treated thoroughly in Section 5. The final Section 6 summarises our results. Throughout, F is a finite field of size q and characteristic p , and G ≤ GL(n, F), n> 1. The natural (right) FG -module of dimension n is denoted V . Whenever we refer to “primitive” or “imprimitive” linear groups, we are implicitly assuming them to be irreducible. 2. Abelian linear groups Suppose A is an abelian irreducible subgroup of GL(n, F), and denote by Δ the enveloping algebra A F of A . Proposition 2.1. Δ is a field, |Δ: F1 n | = n , so that A is cyclic of order dividing q n - 1 . Hence Δ × is a maximal abelian subgroup of GL(n, F) . Proof. The first two claims are covered by [8, Theorem 4, p.99]. The rest is then clear.  Corollary 2.2. A subgroup C of Δ × is irreducible if and only if |C  F : F1 n | = n . There always exist elements of GL(n, F) of order q n - 1, and if g is any such element then g is irreducible. A cyclic subgroup of GL(n, F) of order q n - 1 is called a Singer cycle. The next few results about Singer cycles are well-known. Proposition 2.3. A subgroup C of Δ × is irreducible if and only if |C | does not divide q m - 1 for any proper divisor m of n . Proposition 2.4. Let C be an irreducible subgroup of Δ × . The GL(n, F) -centraliser of C is Δ × , and the GL(n, F) -normaliser of C is the semidirect product of Δ × with a cyclic group of order n . 1