Rank-Engel conditions on linear groups

Abstract

We study linear groups G for which for every g in G there exists a subgroup E of G satisfying some sort of rank condition and such that for every x in G there is a positive integer m such that for all n ≥ m the repeated commutator [x, _ng] lies in E. If E can always be chosen to be a Chernikov group (resp. a polycyclic-by-finite group) such G can be completely described (Wehrfritz in Adv Group Theory Appl 7:143–157, 2019; Wehrfritz in Boll Unione Mat Ital, to appear). For more general rank conditions our analyses below are complete for positive characteristics but are less so for characteristic zero.

If g is an element and H a subgroup of a group G, adapting slightly terminology from [8], call a subgroup E of H a sink for g (or a g-sink) in H if for each h in H there exists a positive integer m(h) such that [h, _ng] ∈ E for all n ≥ m(h). If X is a class of groups, say that a group G is X-Engel if every g in G has a sink in G belonging to X.

There are many different rank conditions on groups, but by Lemma 1 below for linear groups, apart from the Chernikov groups and the polycyclic-by-finite groups, the only classes of interest for the Engel-type conditions here are just three (four in positive characteristic). Below X_fr denotes the class of all groups of finite (Prüfer) rank, X_ftr the class of all soluble-by-finite groups of finite abelian total rank (see below) and X_mm the class of all minimax groups, i.e. poly (groups that satisfy the minimal-by-maximal condition). Also let X_fh denote the class of all groups with finite Hirsch number; that is all groups G with a series of finite length each factor of which is either locally finite or infinite cyclic, the number of infinite cyclic factors in such a series being the Hirsch number h(G) of G.

Further let F denote the class of finite groups, Ch the class of Chernikov groups and P the class of polycyclic groups, so PF denotes the class of polycyclic-by-finite groups. Also LF denotes the class of locally finite groups; that is, the class of all groups G with h(G) = 0.)

Let G be a linear group. A special case of Shumyatsky’s main theorem in [8] says that G is F-Engel if and only if G is finite-by-hypercentral. By Theorem 1.2 of [14] G is Ch-Engel if and only if G is Chernikov-by-hypercentral and by Theorem 1.4 of [14] if G has positive characteristic, then G is PF-Engel if and only if G is PF-by-hypercentral. PF-Engel linear groups of characteristic zero are much more varied. They are completely characterized in [15]. Among the linear groups G of characteristic zero they are exactly the ones with a normal series <1> = T₀ ≤ T₁ ≤ ··· ≤ T_s = T ≤ G with s and the index (G : T) finite and each T_i/T_i−1 polycyclic-by-finite, or G-hypercentral with [T_i, T] ≤ T_i−1, or G-hypercentral, abelian and Chernikov.

FormalPara Theorem 1

Let G be a subgroup of GL(n, F), where n is a positive integer and F is a field of positive characteristic and let X be one of the five classes X_fr, X_ftr, X_mm, X_fh and LF. Then G is X-Engel if and only if G has a normal X-subgroup N such that G/N is hypercentral.

X_fr is the main case in Theorem 1, the other four cases can be thought of as corollaries of it or its proof. If the group G is linear of characteristic zero, then G ∈ X_fh if and only if G ∈ X_fr, so G is X_fh-Engel if and only G is X_fr-Engel. Thus in characteristic zero we have really only three cases to consider.

FormalPara Theorem 2

Let G be an X-Engel subgroup of GL(n, F), where n is a positive integer, F is a field of characteristic zero and X is X_fr, X_ftr or X_mm. Then G has a normal series

$$ {<}1{>} = {\text{U}}_{0} \le {\text{V}}_{1} \le {\text{U}}_{1} \le \cdots \le {\text{V}}_{\text{t}} \le {\text{U}}_{\text{t}} = {\text{U}} \le {\text{T}} \le {\text{G}} $$

such that t ≤ n(n − 1)/2, G/T is finite, T/U is abelian, U is unipotent and for each i with 1 ≤ i ≤ t, [U_i, U] ≤ U_i−1, U_i/U_i−1 is torsion-free (abelian) and either U_i/U_i−1∈ X with V_i = U_i, or [U_i, T] ≤ U_i−1 with V_i/U_i−1∈ X and U_i/V_i G-hypercentral. Also there is a normal subgroup S of G with U ≤ S, S/U ∈ X_fr (even S/U ∈ X_mm if X = X_mm) and G/S hypercentral. Further if X = X_mm then each T/C_T(U_i/U_i−1) is finitely generated (and abelian).

FormalPara Theorem 3

Let G be a connected (in the Zariski topology) subgroup of GL(n, F), where n is a positive integer and F is a field of characteristic zero. If X is X_fr, X_ftr or X_mm, then G is X-Engel if and only if G has a normal series

$$ {<}1{>} = {\text{U}}_{0} \le {\text{U}}_{1} \le \cdots \le {\text{U}}_{\text{t}} = {\text{U}} \le {\text{G}}, $$

where t is finite, G/U is abelian and for each i we have [U_i, U] ≤ U_i−1 and either U_i/U_i−1 ∈ X or [U_i, G] ≤ U_i−1.

FormalPara Corollary

Let G subgroup of GL(n, F), where n is a positive integer and F is a field of characteristic zero and let X denote X_fr, X_ftr or X_mm. Then G is a finite extension of an X-Engel group if and only if G has a series

$$ {<}1{>} = {\text{U}}_{0} \le {\text{U}}_{1} \le \cdots \le {\text{U}}_{\text{t}} = {\text{U}} \le {\text{T}} \le {\text{G}}, $$

where t is finite, G/T is finite, T/U is abelian and for each i we have U_i normal in T, [U_i, U] ≤ U_i−1 and either U_i/U_i−1 ∈ X or [U_i, T] ≤ U_i−1.

The following simple result complements the LF case of Theorem 1. Its short proof does not depend on any of the theorems above.

FormalPara Theorem 4

Let G subgroup of GL(n, F), where n is a positive integer and F is a field of characteristic zero. Then G is LF-Engel if and only if G is (locally finite)-by-hypercentral.

1 General Lemmas

Let G be a subgroup of GL(n, F), where n is a positive integer and F is a field of characteristic zero. Consider the following.

• (a) G has finite Hirsch number h(G).

• (b) G is an FAR-group (cf. [5]); that is, G is soluble-by-finite with h(G) finite and satisfying min-q, the minimal condition on q-subgroups, for every prime q.

• (c) G has finite rank.

• (d) G is an FATR-group (again cf. [5]); that is, G satisfies (b) and π(G) = {primes q : G has an element of order q} is finite.

• (e) G is minimax.

Lemma 1a

(char F = 0) (a), (b) and (c) are equivalent, (d) implies (c) and (e) implies (d).

Now suppose F has characteristic p > 0 and consider the following further two conditions.

• (aʹ) h(G) is finite and G satisfies min-p.

• (cʹ) G is abelian-by-finite and rankG is finite.

Lemma 1b

(charF > 0). (aʹ), (b), (c) and (cʹ) are equivalent, (d) implies (c) and (e) implies (d).

Thus for linear groups we are effectively reduced to just four rank classes to consider, namely those defined by (a), (c), (d) and (e); that is, the classes X_fr, X_ftr, X_mm and in positive characteristic X_fh. Trivially (a′) implies (a). We denote the maximum periodic normal subgroup of a group G by τ(G).

Proof

Suppose charF = 0. If h(G) is finite, then G/τ(G) has a poly(torsion-free abelian of finite rank) normal subgroup of finite index (e.g. [12] Lemma 4). Also G satisfies min-q for every prime q ([9] 9.1iv)) and τ(G) has an abelian normal subgroup of finite index and rank at most n. Therefore (a) implies (b) and (c). Trivially (b) implies (a). If G has finite rank, the G is soluble-by-finite by a theorem of Platonov ([9] 10.9). Thus (c) implies (a). Trivially (d) implies (b). Let G satisfy (e). Linear groups with the minimal condition are Chernikov ([9] 9.8) and linear groups with the maximal condition are polycyclic-by-finite (by a theorem of Tits, see [9] 10.18). Then [9] 5.11 and 6.4 yield that G is soluble-by-finite. Hence (e) implies (d).

Now assume charF = p > 0. Suppose G is periodic and satisfies min-p. Then the Sylow p-subgroups of G are finite and hence (see [9] 9.7) G is a finite extension of an abelian p′-subgroup of rank at most n. Thus as in the proof of the characteristic zero case we obtain that (a′), (b) and (c) are equivalent. Also (c) and (c′) are equivalent by [9] 10.9. As in the previous case we deduce that d) implies (b) and (e) implies (d). The proof of Lemma 1 is complete.□

If π is any infinite set of primes then the direct product D(π) of cyclic groups of order q one for each q in π satisfies (c) but not (d). The additive group Q of the field Q of rational numbers satisfies (d) but not (e). Also D(π) embeds into GL(1, C). If p ∉ π then D(π) embeds into GL(1, F) where F is the algebraic closure of the field of p elements. Further Q embeds into GL(1, C) (even into GL(2, Q)) and if p > 0 also into GL(1, F) for F any large enough field of characteristic p (see [9] 2.2).

Lemma 2

Let G be an X-Engel subgroup of GL(n, F), where the class X satisfies SX = X (i.e. X is subgroup closed) and does not contain the free group of rank 2. If charF = 0, then G is soluble-by-finite. If charF = p > 0, then G is soluble-by-(locally finite and linear of characteristic p).

Proof

Suppose H = <x, y> ≤ G is free of rank 2 on {x, y}. Then H is X-Engel, so a sink for y in H is either trivial or infinite cyclic. But

$$ \left[ {{\text{x}}, \, _{\text{m}} {\text{y}}} \right]^{\text{h}} = {\text{y}}^{{{-}({\text{m}} - 1)}} {\text{x}}^{ - 1} \ldots {\text{ xy}}^{\text{m}}\quad {\text{for all m}} \ge 1\;\;{\text{and h}} \ge 1. $$

Thus [x, _my]^h ≠ [x, _m+2y]^k for all m ≥ 1, h ≥ 1 and k ∈ Z. Consequently < [x, _my], [x, _m+2y] > is not cyclic and hence neither is < [x, _ny] : n ≥ m > for any m ≥ 1. Therefore no such H exists and the claims of Lemma 2 follow from [9] 10.17 (Tits’ theorem), 5.11 and 6.4.□

Lemma 3

Let A be a module over a group G with A additively a q-group for some prime q. For each i ≥ 1 set A_i = {a ∈ A : qⁱa = 0}. If A₁ is G-hypercentral, then so is A.

Proof

Clearly each A_i is a G-submodule of A, A_i ≤ A_i+1 for each i and A = ⋃_i A_i. Also multiplication by qⁱ determines a G-embedding of A_i+1/A_i into A₁. Thus each A_i+1/A_i is G-hypercentral and consequently so is A.□

Lemma 4

Let G be a subgroup of GL(n, F), where F is an algebraically closed field of positive characteristic. Suppose A is an abelian normal subgroup of G of finite rank. Then there exists a normal subgroup D ≤ A of G with A/D finite and the index (G : C_G(D)) dividing n!.

Proof

Let a = a_ua_d denote the Jordan decomposition of the element a of A (see [9] 7.2) and set A_u = {a_u: a ∈ A} and A_d = {a_d: a ∈ A}. The (Zariski) closure of A in GL(n, F) is abelian ([9] 5.11) and contains A_u and A_d ([9] 7.3). Then A_u and A_d are subgroups of GL(n, F) (see [9] 7.1) that are normalized by G. Also AA_d = A_uA = A_u × A_d.

Now A_u is unipotent and A_d is diagonalizable ([9] 7.1). Further A_u ≅ AA_d/A_d has finite rank, as well as being abelian of finite exponent. Therefore A_u is finite and consequently so is A/(A ∩ A_d) ≅ A_u. Set D = A ∩ A_d. Finally (G : C_G(D)) divides n! by [9] 1.12.□

For any group G denote its upper central series by {ζ_s(G): 0 ≤ s ≤ σ}, s and σ ordinals and its hypercentre by ζ(G) = ⋃_s ζ_s(G). Recall, F denotes the class of finite groups.

Lemma 5

Let X be a class of groups and A ≤ T normal subgroups of a group G with [A, T] = <1> and G/T finite such that for each g in G there is a sink A(g) ∈ X in A. If

• (a) X = SX and whenever U, V ≤ W, where W is an abelian group and U and V lie in X, then UV ∈ X, or

• (b) X = SX = QX = D₀X (Q the quotient operator and D₀ the (direct product of finitely many groups) operator), or

• (c) X = X_ftr,

then there exists a normal X-subgroup B ≤ A of G with A/B ≤ ζ_ω(G/B). If also A = T, then G is XF-by-hypercentral.

Note that X_fr and X_mm both satisfy (b) but X_ftr does not. Also XF = X for X = X_fr, X_ftr or X_mm.

Proof

(a) Let X be a (finite) transversal of T to G. Now for each g in G, a in A and large enough n we have [a, _ng] ∈A(g). Clearly g = by for some b in T and y in X and then A(x)^y = A(x)^g. Set

$$ {\text{B}} = < {\text{A}}\left( {\text{x}} \right)^{\text{g}} :{\text{x}} \in {\text{X}},{\text{g}} \in {\text{G}} > = < {\text{A}}\left( {\text{x}} \right)^{\text{y}} :{\text{x}},{\text{y}} \in {\text{X}} >. $$

Clearly B ≤ A is normal in G and B ∈ X. Also A/B consists of right Engel elements of G/B. Consequently A/B ≤ ζ_ω(G/B) by [9] 8.1.

Suppose A = T. Then (G/B : ζ(G/B)) is finite, so by [3] (or see Theorem C of [13]) there exists N/B, a finite normal subgroup of G/B, such that G/N is hypercentral. Trivially N ∈ XF.

(b) UV ≅ (U × V)/{(u, v): u ∈ U, v ∈V, uv = 1} ∈ QD₀X = X. Now apply a).

(c) By (b) there exists B ≤ A with B normal in G, A/B G-hypercentral and rankB finite. Let P = τ(B). Now for any g in G, A(g)∩P ∈ SX = X, so A(g)∩P is Chernikov and hence so is S₀ = < A(x)∩P : x ∈ X > . There is a finite set π of primes such that S₀ ≤ S = O_π(P). Choose π to contain all the prime divisors of (G : T). Note that S is also Chernikov. Further P/S consists of right Engel elements of G/S and C_G(P) ≥ T has finite index in G. Therefore P/S is G-hypercentral by [9] 8.1 again.

Pick U ≤ B maximal subject to P∩U = S. Then B/U is a (periodic) π′-group and hence so is B/U_G for U_G = ⋂_g∈G U^g = ⋂_x∈X U^x. Clearly P∩U_G = S. Now choose V ≤ B maximal subject to U_G ≤ V ≤ B, V normal in G and P∩V = S. Clearly B/V is a π′-group. Let q ∈ π′ and set B_q = {b ∈ B: b^q ∈ V}. Then B_q/V is a finite (recall rankB is finite) F_qG-module. Moreover T ≤ C_G(B) and G/T is a finite π-group. Therefore B_q/V is completely reducible as F_qG-module (Maschke’s theorem). If I/V is an irreducible F_qG-submodule of B_q/V, then P∩I > S = P∩V, so (P∩I)V = I. Hence B_q ≤ PV. Also PV/V ≅ P/S is G-hypercentral. Consequently B/V is G-hypercentral by Lemma 3. We chose B with A/B G-hypercentral. Hence A/V is G-hypercentral. Also P∩V = S, which is Chernikov and V/(P∩V) ≅ VP/P ≤ B/P, which is torsion-free abelian of finite rank. Therefore V ∈ X_ftr.

Finally suppose A = T. Then (G/V: ζ(G/V)) is finite. Hence there exists ([3] again) N/V, a finite normal subgroup of G/V with G/N hypercentral. Clearly N ∈ X_ftr. The proof of Lemma 5 is complete.□

Lemma 6

Let G be a group with G/ζ(G) locally finite. Then there exists a locally finite normal subgroup K of G with G/K hypercentral.

Proof

Let X be a finitely generated subgroup of G. Then X/ζ(X) is finite, so there exists N_X a finite normal subgroup of X with X/N_X hypercentral ([3] again). Choose N_X of least order. If Y is a finitely generated subgroup of G containing X, then X∩N_Y ≥ N_X. Set N = ⋃_X N_X. Then N is a locally finite normal subgroup of G with G/N locally nilpotent.

Define K by K/N = τ(G/N). Then K is also locally finite and H = G/K is locally nilpotent and torsion-free with H/ζ(H) locally finite. Now ζ₁(H) is isolated in H, see [5] 2.3.8, so H/ζ₁(H) is torsion-free. A simple induction yields that H/ζ(H) is also torsion-free. But H/ζ(H) is locally finite. Therefore H = ζ(H); that is, G/K is hypercentral.□

2 Positive characteristic case

The proposition below completes the proof of Theorem 1.

Proposition 1

Let G be an X-Engel subgroup of GL(n, F), where n is a positive integer and F is a field of positive characteristic p. Suppose one of the following holds.

• (i) X is a class of groups of finite rank satisfying X = SX = QX = D₀X = XF.

• (ii) X = X_ftr.

• (iii) X = X_fh.

• (iv) X = LF.

Then G is X-by-hypercentral.

Proof

(i) We make heavy use in this case of X = QX, which is why we have to handle (ii) separately. We mirror the proof of Theorem S in [14].

(a) G is soluble-by-finite.

If this is false, then using Lemma 2 we may assume that G is infinite, periodic and simple, and then that G is of Lie type. Then, using [1] 6.3.1, we may assume that G is actually PSL(2, k) for some locally finite, infinite field k of characteristic p. Consider SL(2, k)∩Tr(2, k), which is isomorphic to the split extension H = k*k⁺, where k⁺ denotes the additive and k* the multiplicative group of k and k* acts on k⁺ via squares (i.e. b^a = ba² for a ∈ k* and b ∈ k⁺).

We show that H modulo <− 1> ≤ k* is not X-Engel, which will show that PSL(2, k) is not X-Engel either and will complete the proof of (a). For if otherwise, then h ∈ H would have a sink E ∈ X in H with E ≤ k⁺. But E has finite rank and k⁺ is elementary abelian. Thus E is finite, H is finite-Engel and H is finite-by-hypercentral (Shumyatsky’s theorem, [8] or [14]). Then (H : ζ(H)) is finite by the theorem of [2]. But if a ∈ k*\<− 1> and b ∈ k⁺\<0>, then b^a = ba² ≠ b. Hence k⁺∩ζ₁(H) = <0>, k⁺∩ζ(H) = <0> and k is finite, which is false. Claim a) is now proved.

(b) If G ≤ Tr(n, F), then G is finite-by-nilpotent.

For if E is an X-subgroup of G′ ≤ Tr₁(n, F), then E has finite rank as well as being nilpotent of finite exponent. Consequently E is finite, G is finite-Engel and (b) of the proof of Theorem S of [14] applies directly.

(c) G is nilpotent-by-finite.

(c) Follows almost immediately from (b). In fact, cf. the proof of (c) in [14] Theorem 1.1, the connected component G^o of G containing 1 is nilpotent.

(d) G is X-by-hypercentral.

Let N be a nilpotent normal subgroup of G of finite index. We may reduce to the two cases where N is unipotent or N is a d-group, see Remark 3.3 of [14].

If N is unipotent, since X-groups have finite rank, every X-subgroup of G is finite. Consequently G is finite-Engel, finite-by-hypercentral and therefore X-by hypercentral (recall F ⊆ X). Suppose N is a d-group. Then N is abelian-by-finite by [9] 7.7 and 3.5 and consequently so is G. Here XF = X. Therefore G is X-by-hypercentral by Lemma 5. The proof of Part (i) is now complete.

(ii) Here X is X_ftr. By Part (i) there is a normal subgroup X of G of finite rank with G/X hypercentral. Using Lemma 1 it follows that X^o is abelian, normal in G and of finite index in X. By Lemma 4 there exists a normal subgroup Y ≤ X^o of G with (X : Y) finite and (G : C_G(Y)) dividing n!. By Lemma 5 there is a normal X_ftr-subgroup Z ≤ Y of G with Y/Z ≤ ζ(G/Z). Now X/Y is finite and G/X is hypercentral. Hence (G/Y : ζ(G/Y)) is finite by the theorem of [2]. But then (G/Z : ζ(G/Z)) is also finite, so by [3] there exists N/Z a finite normal subgroup of G/Z with G/N hypercentral. Finally Z and hence N lie in X_ftr.

(iii) Here X is X_fh. By Lemma 2 there is a soluble normal subgroup N of G with G/N locally finite. We may choose N triangularizable (over an extension of F) and closed in G. Then O_p(N) is also closed in G and hence H = G/O_p(N) is isomorphic to a linear group of characteristic p, see [9] 6.4, such that the image M of N is abelian and contains no non-trivial unipotent (i.e. p-) elements. Hence there is a linear representation of H faithful on M and such that the image of M is diagonalizable (see Remark 3.3 of [14]). Then 1.12 of [9] yields that (G: C_G(N/O_p(N)) = (H:C_H(M)) is finite. Set T = τ(N) ≥ O_p(N). Then N/T is torsion-free abelian, so any X_fh-subgroup of N/T lies in X_fr. By Lemma 5 there exists S a normal subgroup of G with T ≤ S ≤ N, with S/T ∈ X_fr and with N/S ≤ ζ(G/S). By Lemma 6 applied to G/S there is a normal subgroup K of G with S ≤ K, with K/S locally finite and G/K hypercentral. Clearly K ∈ X_fh. This completes the proof of (iii).

(iv) Here X is LF. Set T = τ(G) ≥ O_p(G). Now O_p(G) is closed in G, so G/O_p(G) is isomorphic to some linear group of characteristic p ([9] 6.4). Therefore G/T is also isomorphic to a linear group of characteristic p, see [10]. Consequently we may assume that T = <1>. Certainly G is X_fh-Engel so by iii) there is a normal subgroup N of G with G/N hypercentral and N ∈ X_fh. Also τ(N) ≤ T = <1>, so N is soluble-by-finite with O_p(N) = <1>. Hence N contains a normal subgroup M of G with N/M finite and M torsion-free abelian.

Now G is LF-Engel. If g ∈ G there exists a locally finite subgroup of M that is a g-sink in M and yet M is torsion-free. Hence M consists of right Engel elements of G and consequently M ≤ ζ(G) by [9] 8.1. Finally G/M is finite-by-hypercentral, so (G/M : ζ(G/M)) is finite by [2]. But then (G : ζ(G)) is finite, so there exists a finite normal subgroup K of G with G/K hypercentral by [3]. Consequently G is LF-by-hypercentral. (Actually here K = <1> since T = <1> and G itself is hypercentral.) The proof of Proposition 1 is now complete.□

3 Characteristic zero case

Lemma 7

Let F be a field of characteristic 0, A a non-trivial subgroup of the multiplicative group F* of F and B a non-zero subgroup of the additive group F⁺ of F such that BA = B. Suppose the split extension G of B by A is X_fr-Engel. Then B has finite rank. If G is actually X_mm-Engel then A is finitely generated and B is minimax.

Note that B is torsion-free abelian, so total-rankB = rankB. In particular the G in Lemma 7 is X_fr-Engel if and only if G is X_ftr-Engel.

Proof

Choose a ∈ A\<1>. By hypothesis there is a subgroup E of B of finite rank such that for all b in B there exists an integer m(b) ≥ 1 such that for all n ≥ m(b) we have b(a − 1)ⁿ ∈ E. Subject to this we choose our E of least possible rank. Set E₁ = <b(a − 1)ⁿ : n ≥ m(b), b ∈ B > ≤ E. Then E₁ is also of finite rank (even minimax if E is minimax), so we may assume that E = E₁. Clearly now E(a − 1) ≤ E.

Since a ≠ 1, so a − 1 ∈ F* is an (additive) automorphism of F⁺. Thus E(a − 1) ≅ E and

$$ \cdots \ge {\text{E}}\left( {{\text{a}}{-} 1} \right)^{{{-}{\text{i}}}} \ge \cdots \ge {\text{E}}\left( {{\text{a}}{-} 1} \right)^{ - 1} \ge {\text{E}} \ge {\text{E}}\left( {{\text{a}}{-}1} \right) \ge \cdots \ge {\text{E}}\left( {{\text{a}}{-}1} \right)^{\text{i}} \ge \cdots $$

where i runs over the positive integers. Set E₂ = ⋃_i≥o E(a − 1)⁻ⁱ. If b ∈ B, then for some n ≥ 1 we have b(a − 1)ⁿ ∈ E, so b ∈ E(a − 1)⁻ⁿ. Therefore B ≤ E₂. Clearly E₂ and B have finite rank, in fact rank equal to the rank of E.

(Further E(a − 1) ≤ E yields that Ea ≤ E. Also BA = B, so E₃ = E <a> = ⋃_i≥0 Ea⁻ⁱ ≤ B. Suppose b ∈ B with ba ∈ E. There exists n ≥ 1 with b(a − 1)ⁿ ∈ E. Then (− 1)ⁿb ∈ E. For any b ∈ E₃ we have ba^m ∈ E for some m ≥ 1, so (ba^m−1)a ∈ E and thus ba^m−1 ∈ E. A trivial induction yields that b ∈ E. This shows that E₃ = E and hence E is normalized by <a>. It follows that every a-sink in <a>B contains a normal such a-sink.)

Since we chose E of least rank, so E/E(a − 1) periodic. But

$$ {\text{E}}\left( {{\text{a}}{-} 1} \right)^{{{-}{\text{i}}}} /{\text{E}}\left( {{\text{a}}{-}1} \right)^{{{-}{\text{i}} + 1}} $$

is isomorphic to E/E(a − 1) for each i. Therefore E₂/E is a periodic. Now assume E is minimax. Then E/E(a − 1) is a Chernikov π-group for some finite set π of primes. It follows that B/E is a π-group of finite rank and therefore B/E is Chernikov. Consequently B is minimax.

Finally it remains to prove in the minimax case that A is finitely generated. Let J be the subring of F generated by A. Then J is a domain and B is a faithful J-module. Also B is torsion-free and, by the above, minimax. Hence B has a free abelian subgroup B₀ of rank r = rankB with B/B₀ a periodic π-group for some finite set π of primes. Let t denote the product of all the elements of π. Then the tensor product Z[t⁻¹]B of Z[t⁻¹] by B over Z is equal to Z[t⁻¹]B₀ and hence is isomorphic to Z[t⁻¹]^(r), the free Z[t⁻¹]-module of rank r. Therefore J embeds into the endomorphism ring of this module and hence into the matrix ring Z[t⁻¹]^r′, where r′ denotes r × r.

There exists a finitely generated subring R ≥ Z[t⁻¹] of the complex numbers, an element x of GL(r, R) and a subgroup A₀ of A of finite index such that (A₀)^x ≤ Tr(r, R). (We now regard J ⊇ A as a subring of R^r′ via the obvious embeddings J ≤ Z[t⁻¹]^r′ ≤ R^r′.) If a ∈ A₀ with a^x unipotent, then (a^x − 1)^r = 0 in R^r′ and hence (a − 1)^r = 0 in the domain J. Consequently a = 1 and the unipotent radical of (A₀)^x is trivial. It follows that A₀ is isomorphic to a subgroup of the full diagonal group D(r, R) and the latter is isomorphic to the direct product of r copies of the group U of units of R. But R is a finitely generated domain so U is finitely generated (e.g. [7]). Therefore A₀ and A are also finitely generated. This completes the proof of Lemma 7.

Lemma 8

Let X ≤ Y ≤ T be subgroups of the group G with X normal in T, T normal in G and G/T finite. Set X_G = ⋂_g∈G X^g and similarly for Y_G. If Y/X lies in some class X of groups, then Y_G/X_G lies in SD₀X. If also Y is normal in T and if Y is any class of groups containing T/C_T(Y/X), then T/C_T(Y_G/X_G) ∈ SD₀QY.

Our only application here of the second part of Lemma 8 is where Y is the class of finitely generated abelian groups.

Proof

Clearly X_G ≤ X∩Y_G, so X_G ≤ (X∩Y_G)_G ≤ X_G and X_G = (X∩Y_G)_G. Now Y/X ∈ X implies XY_G/X ∈ SX. Hence Y_G/(X^g∩Y_G) ≅ X^gY_G/X^g ≅ XY_G/X ∈ SX for all g in G. Therefore

$$ {\text{Y}}_{\text{G}} /{\text{X}}_{\text{G}} = {\text{Y}}_{\text{G}} /({\text{X}} \cap {\text{Y}}_{\text{G}} )_{\text{G}} = {\text{Y}}_{\text{G}} / \cap_{\text{g}} ({\text{X}}^{\text{g}} \cap {\text{Y}}_{\text{G}} ) \in {\text{SD}}_{0} {\text{S}}{\mathbf{X}} = {\text{SD}}_{0} {\mathbf{X}}, $$

using that (G : N_G(X)) ≤ (G : T), which is finite.

Suppose Y is normal in T. Since C_T(XY_G/X) ≥ C_T(Y/X) we have that

$$ {\text{T}}/{\text{C}}_{\text{T}} ({\text{Y}}_{\text{G}} /({\text{X}}^{\text{g}} \cap {\text{Y}}_{\text{G}} )) \cong {\text{T}}/{\text{C}}_{\text{T}} ({\text{Y}}_{\text{G}} /({\text{X}} \cap {\text{Y}}_{\text{G}} )) = {\text{T}}/{\text{C}}_{\text{T}} \left( {{\text{XY}}_{\text{G}} /{\text{X}}} \right) \in {\text{Q}}{\mathbf{Y}}. $$

Also C_T(Y_G/X_G) = ⋂_g C_T(Y_G/(X^g∩Y_G)), so T/C_T(Y_G/X_G) ∈ SD₀QY.□

Proposition 2

Let G be an X-Engel subgroup of GL(n, F), where n is a positive integer, F is a field of characteristic zero and X is X_fr, X_ftr or X_mm. Then G has a normal series

$$ < 1 > = {\text{U}}_{0} \le {\text{V}}_{1} \le {\text{U}}_{1} \le \cdots \le {\text{V}}_{\text{t}} \le {\text{U}}_{\text{t}} = {\text{U}} \le {\text{T}} \le {\text{G}} $$

such that t ≤ n(n − 1)/2, G/T is finite, T/U is abelian, U is unipotent and for each i with 1 ≤ i ≤ t, [U_i, U] ≤ U_i−1, U_i/U_i−1 is torsion-free (abelian) and either U_i/U_i−1 ∈ X with V_i = U_i, or, [U_i, T] ≤ U_i−1 with V_i/U_i−1 ∈ X and U_i/V_i G-hypercentral. Also there is a normal subgroup S of G with U ≤ S, S/U ∈ X_fr (even S/U ∈ X_mm if X = X_mm) and G/S hypercentral. Further if X = X_mm then T/C_T(U_i/U_i−1) is finitely generated (and abelian) for all i. Also T can be chosen to be triangularizable (over some extension field of F) and any triangularizable normal subgroup of G of finite index (e.g. the connected component G^o of G) can be chosen to be T.

Proof

Now G is soluble-by-finite by Lemma 2, so there is a triangularizable (over some extension field of F) normal subgroup T of G of finite index and we can choose T to be any one of these, e.g. T = G^o. Set U = u(T), the unipotent radical of T, so U is normal in G and T/U is abelian.

There is a normal series < 1> = U₀ ≤ U₁ ≤ ··· ≤ U_t = U of T such that for each i ≥ 1 we have [U_i, U] ≤ U_i−1 with Lemma 7 applicable to each U_i/U_i−1 and T/C_T(U_i/U_i−1). Thus for each such i

$$ \begin{aligned} \left( * \right)\;\;\left[ {{\text{U}}_{\text{i}} ,{\text{U}}} \right] & \le {\text{U}}_{{{\text{i}}{-}1}} ,{\text{U}}_{\text{i}} /{\text{U}}_{{{\text{i}}{-}1}} \;{\text{is torsion-free abelian and}} \\ & {\text{either U}}_{\text{i}} /{\text{U}}_{{{\text{i}}{-}1}} \in {\mathbf{X}}\;{\text{or }}\left[ {{\text{U}}_{\text{i}} , {\text{T}}} \right] \le {\text{U}}_{{{\text{i}}{-}1}} . \\ \end{aligned} $$

Since U_i−1 ≥ [U_i, U], so U_i−1 ≥ [(U_i)_G, U] and the latter is normal in G. Consequently [(U_i)_G, U] ≤ (U_i−1)_G. In the same way if [U_i, T] ≤ U_i−1 then [(U_i)_G, T] ≤ (U_i−1)_G. By Lemma 8 (U_i)_G/(U_i−1)_G is torsion-free abelian. Similarly if U_i/U_i−1 ∈ X, then (U_i)_G/(U_i−1)_G ∈ X. Thus the series {(U_i)_G, i = 1, 2, …} also satisfies (*). Further if X = X_mm and [U_i, T] is not contained in U_i−1, then T/C_T(U_i/U_i−1) is finitely generated and abelian (since T/U is abelian). It follows that T/C_T((U_i)_G/(U_i−1)_G) is also finitely generated and abelian (Lemma 8 again). Hence to simplify notation assume we have already chosen the U_i to satisfy (*) with all U_i normal in G.

If U_i/U_i−1 ∈ X, set V_i = U_i. Suppose [U_i, T] ≤ U_i−1. If X = X_fr or X_mm, then G/U_i−1 is X-Engel and so by Lemma 5 there exists a normal subgroup V_i of G with U_i−1 ≤ V_i ≤ U_i, with V_i/U_i−1 ∈ X and with U_i/V_i G-hypercentral. Let X = X_ftr. Now a torsion-free abelian group in X_fr is also in X_ftr, X_ftr ⊆ X_fr and U_i/U_i−1 is torsion-free abelian. Thus by the X_fr case there exists a normal subgroup V_i of G with U_i−1 ≤ V_i ≤ U_i, with V_i/U_i−1 ∈ X and with U_i/V_i G-hypercentral. Finally if X is X_fr or X_mm we can also apply Lemma 5 to T/U ≤ G/U. Hence in these cases (and using [2] and [3] again) there is a normal X-subgroup S/U of G/U with G/S hypercentral. The proof of the proposition is complete (since X_ftr ⊆ X_fr).□

Remark

Note that Theorem 2 follows from Proposition 2. For the proof of Lemma 7 all we need of the class X is that X = SX ⊆ X_fr and that whenever B a torsion-free abelian group, E an X-subgroup of B and ϕ is an monomorphism of B into itself with Eϕ ≤ E, then ⋃_i≥1 Eϕ⁻ⁱ ∈ X. Note that the latter condition holds if also X_fr ∩ LX = X. Further the proof of Proposition 2 is valid if X satisfies these conditions for Lemma 7 to hold and also satisfies X = QX = PX = XF.

For any group G denote the lower central series of G by {γⁱG: i ≥ 1}.

Lemma 9

Let X be a class of groups which contains every homomorphic image of the tensor product of any two abelian X-groups. If G is a group with a central subgroup Z such that G/G′Z ∈ X, then γⁱG/γⁱ⁺¹G ∈ X for every integer i ≥ 2. In particular if G is nilpotent, then G′ ∈ PX.

Proof

If Z = <1>, this is 2.26 of [6] and a very small modification of its proof (or alternatively of the proof of b) on Page 10 of [11]) will produce a proof of Lemma 9.□

Lemma 10

Let A₁ and A₂ be abelian groups and denote the tensor product of A₁ and A₂ by T.

• (i) If each A_i can be generated by r_i < ∞ elements then T can be generated by at most r₁r₂ elements.

• (ii) If rankA_i = r_i for each i, then rankT ≤ r₁r₂.

• (iii) If A₁ and A₂ are minimax, then so is T.

Proof

(i) is obvious and (ii) follows easily from it. For (iii) if each A_i is minimax, there are finitely generated subgroups B_i of A_i with C_i = A_i/B_i divisible, abelian and Chernikov ([6] 10.31). The tensor product of C₁ and C₂ is <0> by (D) of Section 59 of [4] and the tensor product T_ij of A_i and B_j is minimax, being a direct sum of finitely many copies of images of A_i, ibid. (H) and (I). Thus the tensor product of the exact sequences

$$ {\text{B}}_{\text{i}} \to {\text{A}}_{\text{i}} \to {\text{C}}_{\text{i}} \to 0,\quad {\text{i}} = 1,2 $$

([4] 60.3) yields that T is an image of T₁₂ ⊕ T₂₁ and as such is minimax.□

Lemma 11

Let N be a subgroup of the group G and g an element of G with N^g ≤ N. Set M = N^<g>. If N has finite rank, then so does M. If N is soluble and minimax, then so is M.

Proof

Clearly g⁻⁽ⁱ⁺¹⁾Ngⁱ⁺¹ ≤ g⁻ⁱ Ngⁱ for each integer i, so M = ⋃_i g⁻ⁱNgⁱ. If X is a finitely generated subgroup of M, then X ≅ g⁻ⁱXgⁱ ≤ N for some i, so rankM = rankN. Suppose N is soluble and minimax. Clearly then M is soluble (with the same derived length as N).

Suppose M is abelian. Now N^g ≅ N, N^g ≤ N and N is abelian and minimax. Hence N/N^g is periodic and Chernikov. In particular N/N^g is a π-group for some finite set π of primes. Clearly g⁻ⁱNgⁱ/g⁻⁽ⁱ⁺¹⁾Ngⁱ⁺¹ ≅ N/N^g for each i. Consequently M/N is a π-group. It also has finite rank, since M does by the first part. Therefore M/N is Chernikov. Also N is minimax. Hence so is M. In general

$$ {\text{M}}^{{\prime }} = \cup_{\text{i}} \left( {{\text{g}}^{{{-}{\text{i}}}} {\text{Ng}}^{\text{i}} } \right)^{\prime } = \cup_{\text{i}} {\text{g}}^{{{-}{\text{i}}}} {\text{N}}^{{\prime }} {\text{g}}^{\text{i}} \;\;{\text{and}}\;\;{\text{g}}^{ - 1} {\text{N}}^{{\prime }} {\text{g}} = \left( {{\text{N}}^{\text{g}} } \right)^{{\prime }} \le {\text{N}}^{{\prime }} . $$

By induction on the derived length of M we may assume that M′ is minimax and by the abelian case above M/M′ is minimax. Consequently M is minimax.□

Proposition 3

Let <1> = U₀ ≤ U₁ ≤ ··· ≤ U_t = U ≤ T be a normal series of the group T with t finite, T/U abelian and for all i we have [U_i, U] ≤ U_i−1 and either U_i/U_i−1 ∈ X or [U_i, T] ≤ U_i−1, for X some class of groups.

• (i) If X is X_fr or X_mm, then T is X-Engel.

• (ii) If U is torsion-free and X = X_ftr, then again T is X-Engel.

Proof

(i) Now T/U is abelian and hence Engel. Thus by induction on t we may assume that T/U₁ is X-Engel. Since PX = X, if U₁ ∈ X, then T is also X-Engel. Henceforth assume that [U₁, T] = <1>.

Let g ∈ T. There exists a g-sink X/U₁ ∈ X in T/U₁. Clearly we may pick X in U. For all x in T there exists m(x) with [x, _ng] ∈ X for all n ≥ m(x). We may assume that X = <[x, _ng]: x ∈ T, n ≥ m(x)> U₁. Since [x, _ng]^g = [x, _ng][x, _n+1g] we have X^g ≤ X. Set Y = X^<g>. Then U₁ ≤ Y ≤ U, U is nilpotent and Y/U₁ ∈ X by Lemma 11. Clearly Y/U₁ is also a g-sink in T/U₁.

Now Y′ ∈ X by Lemmas 9 and 10. Suppose Y′ = <1>. Then [Y, g] ≅ Y/C_Y(g) and C_Y(g) ≥ U₁. Hence [Y, g] ∈ X and clearly it is a g-sink in Y. In general this shows that [Y, g]Y′ is a g-sink in Y and hence in G. Further [Y, g]Y′ lies in X since PX = X. This completes the proof of (i).

(ii) Here X = X_ftr ⊆ X_fr, so if g ∈ T there is a g-sink X in T with U ≥ X ∈ X_fr. But then X is torsion-free (as U here is by hypothesis) and nilpotent, so its upper central factors are torsion-free of finite rank and hence X ∈ X_ftr. Consequently T is X-Engel.□

Proof of Theorem 3

If G in Theorem 3 is X-Engel, apply Proposition 2 with T taken to be G itself. Conversely if G has a series as in Theorem 3, then G is soluble, connected and so triangularizable. Then G′ is unipotent and hence torsion-free. Replacing each U_i by G′∩U_i produces a series with the same properties as given in the theorem but now with U torsion-free (note that X is subgroup closed). Thus G is X-Engel by Proposition 3.□

Proof of the Corollary

If G has a normal X-Engel subgroup T of finite index, then T^o is a connected normal X-Engel subgroup of G of finite index. Now T^o has a normal series as in Theorem 3, which gives a series for G of the required type. If G has a series as in the Corollary, then T^o is X-Engel by Theorem 3 and G is (X-Engel)-by-finite.□

Proof of Theorem 4

First suppose Q is a torsion-free normal subgroup of an LF-Engel group R. If g ∈ R there is a g-sink X ∈ LF in R. If x ∈ Q there exists m ≥ 1 with [x, _mg] ∈X∩Q = <1>. Therefore Q consists of right Engel elements of R. In particular if either R is linear or Q is abelian with R/C_R(Q) finite, then Q ≤ ζ(R), see [9] 8.15 and 8.1.

Suppose the group G in the theorem is LF-Engel. Then G is soluble-by-finite by Lemma 2. Hence G has a triangularizable normal subgroup T of finite index. By the previous paragraph u(T) ≤ ζ(G) and T/u(T) is abelian. Thus T is locally nilpotent. It now suffices to consider just two cases, the case where T is unipotent and the case where T is diagonalizable, see Remark 3.3 of [14].

Set S = τ(T). If T is unipotent, then S = <1> and T ≤ ζ(G) by the above. If T is diagonalizable, then G/C_G(T) is finite ([9] 1.12). Also T/S is abelian and torsion-free. Consequently T/S ≤ ζ(G/S). Thus (G/S : ζ(G/S)) is finite in both cases, so there exists a finite normal subgroup N/S of G/S with G/N hypercental ([3]). Clearly N is locally finite. Conversely any group with such a normal subgroup is LF-Engel.□