Locally Nilpotent Linear Groups

This article examines aspects of the theory of locally nilpotent linear groups. We also present a new classification result for locally nilpotent linear groups over an arbitrary field F. 1. Why Locally Nilpotent Linear Groups? Linear (matrix) groups are a commonly used concrete representation of groups. The first investigations of linear groups were undertaken in the second half of the 19th century, and currently linear group theory is a highly developed branch of group theory. In the past few decades interest in matrix groups has revived and increased, driven partly by the rapid development of computational group theory. Locally nilpotent groups are a generalization of nilpotent groups. Over the years, many structural and classification results for locally nilpotent linear groups have been obtained. Further progress in the study of these groups is possible using computational techniques. Group theoretical algorithms take as input a finite generating set for a group. The celebrated ‘Tits alternative’ states that a finitely generated linear group G either is solvable-by-finite (that is, G contains a normal solvable subgroup of finite index), or G contains a nonabelian free subgroup. For linear groups of the latter type, some basic computational problems, such as membership testing and the conjugacy problem, are undecidable in general. Nilpotent linear groups on the other hand are solvable-by-finite and so are more suitable for computation (note that the class of nilpotent-by-finite linear 2000 Mathematics Subject Classification. Primary 20F19, Secondary 20H20. This publication has emanated from research conducted with the financial support of Science Foundation Ireland. 38 A. S. Detinko and D. L. Flannery groups—which includes all locally nilpotent linear groups—forms an important subclass of This point is underlined by Gromov’s result [5], which implies that a finitely generated group has polynomial growth if and only if it is nilpotent-by-finite: hence, as explained in [1], certain algorithmic efficiency problems can be successfully overcome for locally nilpotent linear groups. Another motivation for further study of locally nilpotent linear groups lies in possible application to abstract group theory concerns. Here an example is recent work [11] on the Carter conjecture for finite groups. Another example is the almost crystallographic groups, which are nilpotent-by-finite and arise naturally as linear groups over Q (see [4, §5.3]). Note that a finitely generated nilpotent group is polycyclic and so isomorphic to a subgroup of GL(n,Z) for some n; hence algorithms for nilpotent subgroups of GL(n,Q) serve as a key step toward algorithms for abstract finitely generated nilpotent groups. 2. Structure of Locally Nilpotent Linear Groups Research into locally nilpotent linear groups relies heavily on knowledge of the structure of such groups. Systematic study of the structure of locally nilpotent linear groups was carried out by D. A. Suprunenko, beginning in the late 1940s [14]. Among other things, Suprunenko classified the maximal locally nilpotent subgroups of GL(n,F) when F is algebraically closed. Various authors extended some of Suprunenko’s results to other fields. In particular, criteria for finiteness of the number of conjugacy classes of maximal locally nilpotent subgroups of GL(n,F), as well as classification of such groups in some partial cases, have been obtained from a detailed description of the structure of locally nilpotent linear groups over an arbitrary field (see e.g. [8]). In the rest of this section we outline some of the most important structural results for locally nilpotent linear groups. A natural point of focus is the maximals, because each locally nilpotent subgroup of GL(n,F) is contained in a maximal locally nilpotent subgroup (by way of contrast, note that a nilpotent subgroup of GL(n,F) may not be contained in a maximal nilpotent subgroup). We proceed via a standard reduction scheme: reducible → completely reducible→ irreducible→ absolutely irreducible→ primitive. Locally Nilpotent Linear Groups 39 2.1. Reducible Locally Nilpotent Linear Groups. We use standard terminology for linear groups, as in [14, 15]. A reducible subgroup G of GL(n,F) is conjugate to a group of block upper triangular matrices, where the diagonal blocks form the irreducible parts of G, which are irreducible representations of G over F in smaller degree. If G is indecomposable locally nilpotent then a stronger statement holds: the irreducible parts of G are pairwise equivalent ([15, p.223, Theorem 2]), so that G is conjugate to a group of block upper triangular matrices   a(g) a12(g) · · · a1k(g) 0 a(g) · · · a2k(g) .. .. . . . .. 0 0 · · · a(g)   , g ∈ G where a(G) = {a(g) | g ∈ G} ≤ GL(m,F) is irreducible locally nilpotent for some m dividing n, and aij(g) ∈ Mat(m,F). If F is a perfect field then aij(g) = cij(g)a(g) where cij(g) ∈ CMat(m,F)(a(G)); that is, G is contained in the direct product of a completely reducible group over F with equivalent irreducible parts, and a unitriangular group over a division algebra. This reduces study of locally nilpotent linear groups to the irreducible case. Another way to obtain a reduction to the completely reducible case is to use the Jordan decomposition. For each g ∈ GL(n,F) there is a unique unipotent matrix gu ∈ GL(n,F) and a unique diagonalizable matrix gd ∈ GL(n,F) such that g = gdgu = gugd (here F is the algebraic closure of F); see [16, p. 91, 7.2]. If F is perfect then gu and gd are both in GL(n,F). Theorem 2.1. ([16, p.97, 7.11] and [15, p.240, Theorem 6]) Let G ≤ GL(n,F) be locally nilpotent, and define Gu = {g ∈ G | g unipotent}, Gd = {g ∈ G | g diagonalizable}. Then Gu and Gd are normal subgroups of G, and 〈Gu, Gd〉 = Gu ×Gd. The theorem implies that if G is a completely reducible locally nilpotent subgroup of GL(n,F) then every subgroup of G is completely reducible, and in particular every element of G is diagonalizable ([16, p.98, 7.12] and [15, p.239, Theorem 5]). Corollary 2.2. ([16, p.98, 7.13]) If G ≤ GL(n,F) is locally nilpotent, and if for each g ∈ G we have gu, gd ∈ G, then G = Gu ×Gd. 40 A. S. Detinko and D. L. Flannery The hypothesis gu, gd ∈ G for all g ∈ G is satisfied if F is finite. This is a partial case of the following. Theorem 2.3. ([12, p. 136, Proposition 3]) Let F be perfect and G be nilpotent. Define Ĝu = {gu | g ∈ G} and Ĝd = {gd | g ∈ G}. Then Ĝu, Ĝd are subgroups of GL(n,F), and G ≤ Ĝu × Ĝd. The group Ĝd defined in Theorem 2.3 is completely reducible over F. 2.2. Irreducible locally nilpotent linear groups. Each irreducible maximal locally nilpotent subgroup of GL(n,F) can be thought of as an absolutely irreducible maximal locally nilpotent subgroup of GL(m,E) for some m dividing n and field E ⊇ F (see [15, p.217, Theorem 4]). This affords a reduction to the absolutely irreducible case, particularly in the classification of irreducible maximal locally nilpotent linear groups. Further investigation is possible in two directions, which are not mutually exclusive: reduction to p-subgroups of PGL(n,F), and reduction to primitive groups. The former is based on the following theorem. Theorem 2.4. ([15, pp.220–221, Theorems 8, 9]) (i) Let n = p1 1 · · · pk k be the prime factorisation of n, where the pi are pairwise distinct primes. If G is a maximal absolutely irreducible locally nilpotent subgroup of GL(n,F) then G = G1 ⊗ · · · ⊗Gk where Gi ≤ GL(pi i ,F) is maximal absolutely irreducible locally nilpotent, 1 ≤ i ≤ k. (ii) Let p be a prime. An absolutely irreducible subgroup G of GL(p,F) containing F×1pa is a maximal absolutely irreducible locally nilpotent subgroup of GL(p,F) if and only if G/F×1pa is a Sylow p-subgroup of PGL(p,F). Irreducible locally nilpotent linear groups are center-by-periodic; in fact, the central quotient of each is a direct product of p-groups (see [13, Corollary 3.2.4]). Except when F is finite or algebraically closed, the description of Sylow p-subgroups of PGL(n,F) is quite different to the description of Sylow p-subgroups of GL(n,F). Sylow p-subgroups of PGL(n,F) were considered in [6], mainly for p > 2. Classifying the Sylow 2-subgroups of PGL(n,F) is difficult. Locally Nilpotent Linear Groups 41 In [6], p-subgroups of PGL(n,F) are handled using the same techniques as for locally nilpotent linear groups, including the reduction to primitives. The reduction to primitives is not so straightforward for locally nilpotent linear groups as it is for some other classes of linear groups, such as solvable groups. To appreciate this disparity, note that an irreducible imprimitive solvable subgroup of GL(n,F) is conjugate to a subgroup of GoT where G is a primitive solvable linear group and T is a transitive solvable permutation group ([15, p.129, Theorem 5]); however, the wreath product of a locally nilpotent linear group and a nilpotent permutation group need not even be (locally) nilpotent. A discussion of techniques for studying nilpotent primitive subgroups G of GL(n,F) can be found in [8, Section 2]. One technique is to use the series G ≥ H ≥ K ≥ 1 where K = [G,G] and H = CG(K). A basic result here is as follows. Theorem 2.5. ([8, Theorem 2]) Let G ≤ GL(p,F) be primitive absolutely irreducible locally nilpotent, p 6= charF. Then K is an abelian p-group, Σ = 〈K〉F is a field, G/H ∼= Gal(Σ/F), [H, H] ≤ F×1pa , and H is an absolutely irreducible primitive class 2 nilpotent subgroup of GL(m, Σ), m = pa/|Σ : F|. The primitive nilpotent subgroups of GL(n,F) for finite fields F have been completely classified, in [3]. The paper [8] also summarizes results and methods for classifying maximal locally nilpotent subgroups of GL(n,F) over an arbitrary field F. Considerable attention has been paid to the problem of determining when the number of GL(n,F)-conjugacy classes of maximal locally nilpotent subgroups of GL(n,F) is finite. Finiteness of that number depends on finiteness of the groups F×/(F×)m, for m dividing n. Groups over an algebraically closed field have been the most intensively studied. Theorem 2.6. ([14, Chapter III] and [15, Chapter VII]) Let F be algebraically closed. (i) Irreducible locally nilpotent subgroups of GL(n,F) exist if and only if charF = 0 or charF does not divide n, in which cases there exists an irreducible nilpotent subgroup of GL(n,F) of nilpotency class l, for each and every nilpotency class l ≥ 1. 42 A. S. Detinko and D. L. Flannery (ii) Irreducible maximal locally nilpotent subgroups of GL(n,F) are monomial and pairwise conjugate. The matrix form of the groups in (ii) is given in [14, Chapter III, §7]. Note that (ii) implies that a completely reducible locally nilpotent linear group over an algebraically closed field is monomial. In summary, locally nilpotent linear groups constitute a wellstudied class of groups, for which a lot of structural information and efficient methods of investigation are known. However the theory still has significant gaps. Most results deal only with absolutely irreducible maximal locally nilpotent subgroups of GL(n,F). Those results do not readily yield analogous results for locally nilpotent linear groups that are not maximal or are not absolutely irreducible (cf. [3]). Complete classifications of locally nilpotent subgroups of GL(n,F) are feasible only by placing restrictions on the field F or the degree n. In the sequel we allow arbitrary fields but restrict the degree. 3. Prime Degree Locally Nilpotent Linear Groups We now give an illustration of how techniques in the theory of locally nilpotent linear groups may be applied to obtain a full classification in that theory. Specifically, in this section we classify the irreducible maximal locally nilpotent subgroups of GL(q,F), where q is prime and F is any field. This classification is in the form of a complete list of GL(q,F)-conjugacy class representatives of the groups, with each listed group defined by a generating set of matrices. Also we provide criteria to decide conjugacy between listed groups. Restricting to prime degree has several advantages: an irreducible subgroup G of GL(q,F) is either abelian or absolutely irreducible, and is either primitive or monomial. Additionally, if G is absolutely irreducible locally nilpotent then lies in a Sylow q-subgroup of PGL(q,F), and Sylow q-subgroups of PGL(q,F) have a simpler description than do Sylow subgroups of PGL(n,F) for composite degree n. Certainly, a partial classification of the irreducible maximal locally nilpotent subgroups of GL(q,F) can be derived from a description of the absolutely irreducible maximal locally nilpotent subgroups of GL(q,F). However, here we propose other methods and give a complete, self-contained result, which can be extended to get a complete classification in prime power degrees q. In particular we give an exact description of the Sylow 2-subgroups of PGL(2,F), Locally Nilpotent Linear Groups 43 omitted by other authors. This is of special importance because (for example) classification of the Sylow 2-subgroups of PGL(n,F) in arbitrary degree n depends on classification in degree 2 (cf. the case of 2-subgroups of GL(n,F) in [7, 10]). The methods used in this section were originally developed to classify the maximal irreducible periodic subgroups of PGL(q,F), in [2]. We begin by considering absolutely irreducible groups; abelian groups will be treated at the end. By [15, Theorem 6, p. 217], GL(q,F) contains absolutely irreducible locally nilpotent subgroups if and only if F× has an element ξ of order q (of course charF 6= q). Let D = {diag(ββ1, . . . , ββq) | βi ∈ Sylq(F), β ∈ F×}. For α ∈ F×, define Iα = ( 0 1q−1 α 0 ) ∈ GL(q,F) and write I in place of I1. For H ≤ GL(q,F) let Det(H) = {det(h) | h ∈ H}. Assuming ξ ∈ F× we define Hα = 〈D, Iα〉. The subgroup Hα of GL(q,F) is monomial and absolutely irreducible. Since Hα/F×1q is a q-subgroup of PGL(q,F), Hα is locally nilpotent. If Sylq(F) is finite then Hα is nilpotent with nilpotency class 1+(q−1) logq |Sylq(F)|. Denote by π the natural homomorphism from the group of all monomial matrices in GL(q,F) onto the group Sym(q) of q×q permutation matrices. The kernel of π is the group D(q,F) of all diagonal matrices in GL(q,F). Lemma 3.1. (Cf. [2, Lemma 20]) Let a, b ∈ D(q,F). The following statements are equivalent. (i) Ia, Ib are GL(q,F)-conjugate. (ii) Ia, Ib are D(q,F)-conjugate. (iii) det(a) = det(b). Lemma 3.2. Let H be an irreducible monomial locally nilpotent subgroup of GL(q,F). Then H is conjugate in GL(q,F) to a subgroup of Hα for some α ∈ F×. Proof. If H is abelian then π(H) ≤ Sym(q) is a transitive abelian group i.e. a cycle of order q, and H ∩D(q,F) ≤ F×1q. On the other hand if H is absolutely irreducible then HF×/F×1q is a q-group, so that H∩D(q,F) ≤ D, and π(H) is again a cycle of order q. Hence up to conjugacy H ≤ 〈D, Ia〉 for some a ∈ D(q,F). Then H is conjugate to a subgroup of Hdet(a) by Lemma 3.1. ¤ 44 A. S. Detinko and D. L. Flannery Denote the set Det(D) = Sylq(F)(F) by S. Lemma 3.3. If α ∈ S then Hα is D(q,F)-conjugate to H1. If α1, α2 6∈ S then Hα1 and Hα2 are GL(q,F)-conjugate if and only if Det(Hα1) = Det(Hα2) i.e. 〈α1S〉 and 〈α2S〉 are identical subgroups of F×/S of order q. Proof. Suppose α = β1β for some β1 ∈ Sylq(F) and β ∈ F×. Then det(Iα) = det(Ib) where b = diag(β1β, β, . . . , β) ∈ D. Therefore Hα is D(q,F)-conjugate to 〈Ib, D〉 = H1 by Lemma 3.1. Now suppose α1, α2 6∈ S and α1 ∈ 〈α2, S〉. Then det(Iα1) = det(I α2c) for some c ∈ D and 1 ≤ r ≤ q − 1. Also there exists x ∈ Sym(q) such that xI α2cx = Ib for some b ∈ D(q,F). Hence by Lemma 3.1 once more, Hα1 and Hα2 are conjugate (this time by a monomial matrix). ¤ Corollary 3.4. Define H = {Hα | α ∈ F× \ S}. The GL(q,F)conjugacy classes of the groups in H are in one-to-one correspondence with the distinct subgroups of F×/S of order q. Consequently the number of such classes is finite if and only if F×/S is finite. Remark 3.5. If F is algebraically closed or finite then H is empty: a maximal absolutely irreducible monomial locally nilpotent subgroup of GL(q,F) is conjugate to H1. We turn next to primitive groups. Lemma 3.6. Let H be a primitive locally nilpotent subgroup of GL(q,F). Then H has an irreducible abelian normal subgroup. Proof. First we show that H has an abelian noncentral normal subgroup. As H is locally nilpotent, it is solvable, and thus has an abelian normal subgroup A of finite index (see e.g. [15, p.135, Theorem 6]). If A ≤ Z(H) then H/Z(H) is finite and thus H is nilpotent. But a nonabelian nilpotent group certainly contains an abelian noncentral normal subgroup. Any abelian noncentral normal subgroup A of H must be irreducible. For if A were reducible then it would be diagonalizable with inequivalent irreducible parts, which contradicts primitivity of H. ¤ Lemma 3.6 implies that a primitive locally nilpotent subgroup of GL(q,F) is contained in the GL(q,F)-normalizer of the multiplicative group of a field extension ∆ of F1q of degree q. Since this Locally Nilpotent Linear Groups 45 degree is prime, ∆ is a cyclic extension of F, with Galois group of order q. As long as F× has an element ξ of order q, ∆ = 〈h〉F for some h ∈ GL(q,F) such that h = β1q ∈ F×1q by [9, p. 289, Theorem 6.2]. As β ∈ (F×)q implies that h is scalar, we have β = αγ for some α, γ ∈ F× and 1 ≤ r ≤ q−1. Then γ−1h and I α have the same characteristic (minimal) polynomial X − α1q, and because X − α1q is F-irreducible, γ−1h and I α are conjugate. Hence ∆ is conjugate to ∆α := 〈Iα〉F, α 6∈ (F×)q, and NGL(q,F)(∆α ) = 〈∆α , d〉 where d = diag(1, ξ, . . . , ξq−1). Denote by G(α, b) the subgroup 〈Aα, db〉 of 〈∆α , d〉, where Aα ⊇ F×1q, Aα/F×1q is the Sylow q-subgroup of ∆α /F×1q, and b ∈ ∆α . Since Aα is a noncentral irreducible subgroup, G(α, b) is absolutely irreducible. Lemma 3.7. An absolutely irreducible primitive locally nilpotent subgroup H of GL(q,F) is conjugate to a subgroup of some G(α, b). Proof. Up to conjugacy H = 〈H ∩∆α , db〉 for some α ∈ F× \ (F×)q and b ∈ ∆α . Then H ∩∆α ≤ Aα by Theorem 2.4. ¤ We use the notation εk to stand for an element of multiplicative order 2 in the algebraic closure of F. If k = 2 then we drop the subscript; that is, ε is a square root of −1. Lemma 3.8. Suppose F does not have characteristic 2, and ε 6∈ F. Let E = F(ε), and let σ be the F-involution of E. If Syl2(E) = 〈εm〉 is cyclic then Syl2(E/F) is cyclic. Explicitly, one of the following must be true: (i) σ(εm) = −ε−1 m , and Syl2(E/F) = 〈εmF×〉 of order 2m−1; or (ii) σ(εm) = ε−1 m , and Syl2(E/F) = 〈(1+εm)F×〉 of order 2. Proof. We make some preliminary observations. First, either σ(εm)= ε−1 m or σ(εm) = −ε−1 m . Suppose σ(εk) = ε−1 k . Then σ((1+εk) k ) = σ(1+εk) k = (1+ε−1 k ) 2 = ( 1 + εk εk )2k = (1+εk) k . Thus (1 + εk) k ∈ F and so (1 + εk)F× ∈ Syl2(E/F) (1) if k ≥ 2. Also, if k > 2 then ε−1 k (εk−1+1) = ε−1 k (εk +1) = εk +ε−1 k ∈ F× and so 1 + εk−1 ∈ 〈εk〉F×. (2) 46 A. S. Detinko and D. L. Flannery Now let xF× be a nontrivial element of Syl2(E/F) of order 2, meaning that x l ∈ F× \ (F×)2. Suppose l = 1. Write x = a + εb, a, b ∈ F. Then 2abε ∈ F implies a = 0 i.e. x ∈ 〈εF×〉 ≤ 〈εmF×〉. (3) Suppose l ≥ 2. We have σ(x) = yx for some y ∈ Syl2(E), y l = 1. Then y 6= −1, because y = −1 implies |xF×| = 2 < 2. Further, x = σ(x) = σ(yx) = σ(y)yx and so σ(y) = y−1. (4) As tr(x) = (1 + y)x ∈ F, x ∈ (1 + y)−1F× = (1 + σ(y))F× = (1 + y−1)F×. (5) At last we are ready to complete the proof that either (i) or (ii) must be true. Let σ(εm) = −ε−1 m ; then m > 2. By (3), we may take l ≥ 2, in which event 4 < |y| ≤ 2m−1 by (4). Then (1 + y−1)F× ⊆ 〈εm〉F× by (2), and by (5), (i) is proved. If σ(εm) = ε−1 m then by (1), 〈(1 + εm)F×〉 ⊆ Syl2(E/F). Also εmF× ∈ 〈(1 + εm)F×〉2, since σ fixes ε−1 m (1 + εm). Part (ii) now follows from (2), (3), and (5). ¤ Corollary 3.9. If Syl2(E) is quasicyclic then Syl2(E/F) is also quasicyclic, and Syl2(E/F) = {〈εkF×〉 | εk ∈ Syl2(E)}. Proof. For each εk ∈ Syl2(E), σ(εk) = ε−1 k i.e. σ(εk) = −ε−1 k is impossible. The corollary is then a consequence of Lemma 3.8 (ii) and (2) in the proof of the lemma. ¤ Lemma 3.10. Let |E : F| = q and E = F(a), where aql ∈ F. Suppose ξ ∈ F×, and E 6= F(ε) if q = 2. Then Sylq(E/F) = 〈aF×〉. Proof. If q > 2, or q = 2 and ε ∈ F, then the lemma follows from [8, Lemma 2]. Let q = 2 and xF× ∈ E×/F× be of order 2, m ≥ 1, so that x m = α for some α ∈ F× \ (F×)2. If α = −4γ4, γ ∈ F×, then x m−1 /2γ is a square root of −1, contradicting E 6= F(ε). Suppose α 6∈ −4(F×)4. The polynomial X − α is F-irreducible, so that if m ≥ 2 then |E : F| ≥ 4. Therefore m = 1, a = √β for some β ∈ F× and x = √α. For some x1, x2 ∈ F we have √ α = x1 +x2 √ β. Then α = x1 + βx 2 2 + 2x1x2 √ β implies x1 = 0 or x2 = 0; as the latter is impossible we get x ∈ 〈aF×12〉 as required. ¤ Locally Nilpotent Linear Groups 47 Lemma 3.11. If q > 2 or α 6∈ −(F×)2 then Aα is the monomial group 〈Iα,F×1q〉; otherwise, Aα is primitive. Proof. In Lemma 3.10 put E = ∆α and a = Iα. Then Aα = 〈Iα,F×1q〉 unless q = 2 and ∆α ∼= F(ε) i.e. α ∈ −(F×)2. If A−γ2 were monomial then A−γ2 would be in F ×1q; however A−γ2/F×1q contains the element (12 + γ−1I−γ2)F×1q of order 4. ¤ We refer to the set of hypotheses q = 2 and α ∈ −(F×)2 as case (∗). Lemma 3.8 and Corollary 3.9 give an explicit description of the Aα in case (∗). Actually, a group G(−γ2, b) in this case is conjugate to some G(−1, b′), since I−γ2 is D(q,F)-conjugate to γI−1 by Lemma 3.1. In all but case (∗), |G(α, b)/F×1q| = q (because (db) = det(b)1q and [Iα, d ] is scalar) and so G(α, b) is class 2 nilpotent. The group G(−1, b) is locally nilpotent, and it is nilpotent only if Syl2(∆ × −1) is cyclic; then G(−1, b)/F×1q is a dihedral 2-group, and G(−1, b) has nilpotency class log2 |Syl2(∆−1)/F×12|. Lemma 3.12. In case (∗), G(α, b) is primitive. In all other cases, G(α, b) is primitive if and only if det(b) 6∈ 〈(−1)q−1α, (F×)q〉 = Det(Aα). Proof. By Lemma 3.11, assume we are in a case other than (∗). Then [8, Lemma 1] yields that G(α, b) is primitive if and only if all elements of G(α, b) of order q are scalar. Suppose det(b) 6∈ Det(Aα) and let h ∈ G(α, b), |h| = q. If h 6∈ Aα i.e. h = dbb1, b1 ∈ Aα, then h = det(bb1)1q implies that det(b) ∈ Det(Aα). Thus h ∈ Aα, and h is scalar by Lemma 3.11. Conversely, if det(b) ∈ Det(Aα) then for some x ∈ Aα, dbx is a nonscalar element of G(α, b) of order q. ¤ Remark 3.13. Except in case (∗), if F is finite then G(α, b) is monomial. Lemma 3.14. For i = 1, 2, let gi = dbi where bi ∈ ∆α. The following statements are equivalent. (i) g1 and g2 are GL(q,F)-conjugate. (ii) g1 and g2 are ∆α-conjugate. (iii) det(b1) = det(b2). Proof. See [2, Lemma 21]. ¤ Corollary 3.15. Apart from case (∗), primitive groups G(α, b1), G(α, b2) are conjugate if and only if Det(G(α, b1)) = Det(G(α, b2)) and det(b1) = det(b2c) for some c ∈ Aα. 48 A. S. Detinko and D. L. Flannery Proof. Suppose tG(α, b1)t−1 = G(α, b2). Since t normalizes Aα as a consequence of Lemma 3.12, so t ∈ 〈d, ∆α 〉. Then it can be checked that tdb1t−1 ∈ db2Aα. The other direction is clear by Lemma 3.14. ¤ Denote by G the set of all groups G(α, b) that are primitive, subject to the proviso that the only groups in case (∗) included in G are the G(−1, b). Remark 3.16. Note that G is empty if F is algebraically closed, for then G(α, b) is not defined. When F is finite, G is nonempty if and only if q = 2 and |F| ≡ 3 mod 4. Lemma 3.17. If nonempty, the subset G̃ of G consisting of the groups not in case (∗) splits up into finitely many GL(q,F)-conjugacy classes if and only if F×/(F×)q is finite. Proof. If G̃ has only finitely many non-conjugate elements then F×/(F×)q is finitely generated and so finite. Conversely, if F×/(F×)q is finite then there are only finitely many subsets of F× that are possibilities for Det(G(α, b)); hence the number of GL(q,F)-conjugacy classes in G̃ is finite by Corollary 3.15. ¤ The next theorem is our main classification result. Theorem 3.18. Suppose ξ ∈ F×. A subgroup G of GL(q,F) is an absolutely irreducible maximal locally nilpotent subgroup of GL(q,F) if and only if G is conjugate to a group in N = {H1} ∪ H ∪ G, with the following exceptions when q = 2 and ε 6∈ F : (i) H1 is a proper subgroup of G(−1, 1) ∈ G; (ii) if α 6∈ −(F×)2 and either det(b) ∈ −(F×)2 or det(b) ∈ α(F×)2, then G(α, b) is conjugate to a proper subgroup of G(−1, c) where det(c) = α. Proof. We have observed previously that all of the groups in N are absolutely irreducible locally nilpotent. By Lemmas 3.2, 3.3, and 3.7, and remarks after Lemma 3.11, an absolutely irreducible locally nilpotent subgroup of GL(q,F) is conjugate to a subgroup of a group in N . Therefore it remains to show that the Hα and G(α, b) ∈ G are really maximal locally nilpotent, apart from the stated exceptions. Let G be a maximal locally nilpotent subgroup of GL(q,F) containing Hα. If G is monomial then tGt−1 = Hβ for some Hβ and t ∈ GL(q,F). If tDt−1 6= D then tDt−1∩D is scalar of index q in D, Locally Nilpotent Linear Groups 49 so |Hβ/F×1q| = q; but Hβ/F×1q has cardinality at least q. Thus tDt−1 = D, and then q = |Hβ : D| ≥ |tHαt−1 : D| = |Hα : D| = q. Therefore tHαt−1 = Hβ i.e. Hα = G. Now suppose G is primitive, hence conjugate to some G(α1, b). In a case other than (∗) we have |G/F×1q| = q, which is less than the cardinality of Hα/F×1q. Hence q = 2, ε 6∈ F×, G is conjugate to G(−1, b), and Hα = 〈d, Iα,F×12〉. Either Iα or I−α = dIα is conjugate to nonscalar h ∈ A−1 such that h ∈ F×12. Now h has the form γI−1, γ ∈ F×, and by comparing determinants we get α = ±γ2. Thus if Hα ∈ H then Hα is maximal. However H1 = 〈d, I−1,F×12〉 is a proper subgroup of G(−1, 1). Let G be a maximal locally nilpotent subgroup of GL(q,F) containing G(α, b) ∈ G. For some t ∈ GL(q,F), tGt−1 = G(α1, b1) ∈ G. Apart from when q = 2, ε 6∈ F×, and α1 = −1, we have |G(α, b)/F×1q| = |G(α1, b1)/F×1q| = |G/F×1q| = q, and thus G(α, b) = G. Suppose now that q = 2, ε 6∈ F, and α1 = −1. If α ∈ −(F×)2 then G(α, b) is conjugate to some G(−1, b2), so that G(α, b) = G. Therefore if G(α, b) is not maximal then α 6∈ −(F×)2. For the rest of the proof α 6∈ −(F×)2, which means that G(α, b) = 〈Iα, db,F×12〉. One of the following must occur: tIαt−1 6∈ A−1 and tdbt−1 ∈ A−1, or tIαt−1 6∈ A−1 and tdbt−1 6∈ A−1. In the first case, det(b) ∈ −(F×)2. In the second case, the condition that [Iα, db] is scalar forces tdbt−1 ∈ I−1tIαt−1F×, so det(b) ∈ α(F×)2. Suppose det(b) ∈ −(F×)2. Then det(db) = det(γI−1) for some γ ∈ F×. Since also tr(db) = 0 = tr(γI−1), there exists s ∈ GL(2,F) such that sdbs−1 = γI−1. Since IαdbI−1 α = −db, we have that sIαs−1 normalizes ∆−1. Hence sIαs−1 = dc, where c ∈ ∆−1 and det(c) = α. It follows that G(α, b) is conjugate to a proper subgroup of G(−1, c). The reasoning that leads to this same conclusion in the case det(b) ∈ α(F×)2 is entirely similar. ¤ With the next lemma we complete our classification of the irreducible maximal locally nilpotent subgroups of GL(q,F). Lemma 3.19. Suppose GL(q,F) has irreducible abelian subgroups, and let H be an irreducible maximal abelian subgroup of GL(q,F). 50 A. S. Detinko and D. L. Flannery (i) If ξ 6∈ F× then H is a maximal locally nilpotent subgroup of GL(q,F), and moreover any maximal locally nilpotent subgroup of GL(q,F) is abelian. (ii) Let ξ ∈ F×. Then H is a maximal locally nilpotent subgroup of GL(q,F) unless q = 2 and ε 6∈ F. If q = 2 and ε 6∈ F then H is a maximal locally nilpotent subgroup of GL(q,F) if and only if H/F×12 is not a 2-group. Proof. so that if H is not maximal locally nilpotent then q = 2, ε 6∈ F, and H = Aα ≤ G(α, β) ∈ G, by Lemma 3.11. ¤ One situation in which all irreducible maximal locally nilpotent subgroups of GL(q,F) are nonabelian is when q = 2 and |F| is a Mersenne prime. Acknowledgment. This article is part of research carried out in the Department of Mathematics at NUI, Galway over a number of years. We are grateful to Professor Martin Newell for his ongoing generous support and encouragement of our work during this time.