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28 O.D. ARTEMOVYCH Proof. Suppose that G is a non-“nilpotent-by-Černikov” group. Then D is an indecomposable Q〈u〉-module. By Theorem 2.4 of [30] D ∼ = E(Q〈u〉/I), where E(Q〈u〉/I) is an injective envelope of Q〈u〉/I and I is an irreducible ideal of Q〈u〉. By Lemma 7.12 of [31] I is a P -primary ideal for some P ∈ Spec(Q〈u〉) and by Proposition 3.1 of [30] D ∼ = E(Q〈u〉/P ). Let A 1 = {x ∈ D | xP = {0}}. By Theorem 3.4 of [30] A 1 and the field Q〈u〉/P are isomorphic as linear (Q〈u〉/P )-spaces. Since Ann Q〈u〉 (A 1 ) ≠ {0}, by Corollary 7.3 of [28] A 1 is a module direct sum of cyclic Q〈u〉-submodules. In view of Proposition 2.2 of [30] A 1 is a cyclic Q〈u〉-submodule. Hence A 1 ⋊〈u m 〉 is a nilpotent subgroup for some integer m ≥ 2. By Z we denote the centre Z(A 1 ⋊ 〈u m 〉). Without restricting of generality we can assume that Z(A 1 ⋊ 〈u〉) = 1. Since the quotient group A 1 /(A 1 ∩ Z) is torsion-free by Mal’cev Theorem (see [22, Proposition 5.2.19]), the subgroup A 1 ∩Z is pure in A 1 (see [21, §26]). Consequently A 1 ∩ Z is a divisible subgroup of A 1 . Moreover Z is a 〈u〉-invariant subgroup. This means that A 1 ∩ Z is an injective right Q〈u〉-submodule of A 1 and hence A 1 ∩ Z = A 1 . By U we denote the quotient group 〈u〉/〈u m 〉. Let π be a finite set of distinct primes p 1 , . . . , p l (l ≥ 2). By Lemma 2.3 of [16] the nontrivial right ZU-module A 1 has a submodule N such that A 1 /N is a torsion π-group. Clearly that N is a normal in G. Let C = D/N. Then C is a right Q〈u〉-module with the action induced by the conjugation of u on A 1 . Moreover C is a P - module (see e.g.[28, §3]) and by Theorem 9.4 of [28] C has a basic submodule B. If B = C, then G/N is a nilpotent-by-Černikov group, a contradiction. Hence B ≠ C and by Theorem 5.28 of [28] the quotient module C/B is injective. In view of Proposition 2.2 of [30] C/B is a nilpotent-by-Černikov group. By Lemma 2.6 G is a nilpotent-by-Černikov group. Lemma 3.3. Let G = A ⋊ 〈u〉 be the semidirect product of a normal abelian torsion-free subgroup A and an infinite cyclic subgroup 〈u〉. If G satisfies Max-NČ then it is either a polycyclic group or a nilpotent-by-Černikov group. Proof. By Theorem 21.3 of [21] A = D × F is a group direct product of the divisible part D and a reducible subgroup F . We also assume that G is neither a polycyclic group nor a nilpotent-by-Černikov group. Then G is a locally “nilpotent-by-finite” group. (1) First, suppose that the divisible part D is trivial. Let B be a p-basic subgroup of F for some prime p. If B = F is a finitely generated subgroup, then G is polycyclic. Assume that B = F is not finitely generated. Then by Lemma 3.1 the quotient group G/B p is nilpotent-by-Černikov and by Lemma 2.6 so is G, a contradiction. Let B ≠ F . If B is not finitely generated, then by Lemma 26.1 of [21] B = B/B p is a p-pure subgroup of F = F/B p and then by Proposition 27.1 of [21] F = B × C is a group direct product of an infinite abelian subgroup B of prime exponent p and a p-divisible subgroup C. Thus F/F p is an infinite group and by Lemma 3.1 G/F p is a nilpotent-by- Černikov group. In view of Lemma 2.6 G is the ones, a contradiction. Hence B is a finitely generated subgroup. (a) Suppose that the quotient group F/B is torsion and has a non-trivial p-subgroup. Then we can assume that F/B ∼ = C p ∞. Therefore 〈B, u〉 is a nilpotent-by-Černikov subgroup and by Lemma 3.7 of [1, Chapter 3, §5] 〈B, u〉 = N ⋊ 〈u〉 for some G-invariant subgroup N of F . Since 〈B, u〉 ≠ G, we can assume that B = N. Moreover 〈B, u m 〉 is a nilpotent subgroup for some integer m and therefore B has a 〈B, u m 〉-invariant subgroup B 1 with the infinite cyclic quotient group B/B 1 . Let Ĝ1 = (F ⋊ 〈u m 〉)/B 1 = ˆF ⋊ 〈û m 〉. It is clear that ˆF ∼ = Q (p) . Since F ⋊〈u m 〉 is a non-“nilpotent-by-Černikov” subgroup, [F, 〈um 〉] B 1 in view of Proposition 2.3 and consequently û m induces an automorphism of ˆF of infinite order. But
SOLVABLE GROUPS AND NILPOTENT-BY-ČERNIKOV SUBGROUPS 29 as stated in Exercise 5 of [32, §113] Aut Q (p) ∼ = Z2 × Z and therefore Ĝ1 is not a locally “nilpotent-by-finite” group, a contradiction. (b) If F/B is a p ′ -group, then it has an infinite Sylow q-subgroup of finite index for some prime q. Therefore we can assume that F/B is an infinite q-group. If S/B is a basic subgroup of F/B then, in the same manner as before we can prove that S/B is finitely generated and F/S ∼ = C p ∞, a contradiction. (c) Now suppose that the quotient group F/B is mixed. Without restricting of generality we may assume that F is a p-divisible group. Then H = ζ α G · 〈u〉 is a hypercentral group. Assume that F/((ζ α G) ∩ F ) is a finitely generated group. In view of Proposition 2.3 the quotient group H/H ′ is finitely generated. Then H = H ′ E for some finitely generated subgroup E. If H = H/γ c+2 H, where c is the nilpotent class of E, then H = H ′ E = E. This yields that L = [L, E], where L = γ c+2 H, and so H = LE = [L, 〈u〉]E. (**) Since the quotient F/L is a finitely generated abelian p-divisible group, it is finite and consequently L ⋊ 〈u〉 is a subgroup of finite index in G. In view of our hypothesis and Lemma 2.6 L/L p is a finite quotient group for all primes p. By ( ∗∗ ) L is a divisible group. Lemma 3.2 yields a contradiction. If the quotient group F/((ζ α G) ∩ F ) is not finitely generated, then without restricting of generelity we can assume that the centre Z(G) is trivial. Let K be any finitely generated subgroup of F . Then 〈K, u s 〉 is a nilpotent subgroup for some integer s. By X we denote the subgroup Z(〈K, u s 〉)∩F . Then X is a finitely generated 〈u〉-invariant subgroup and moreover X is a non-trivial right ZU-module, where U = 〈u〉/〈u s 〉 and the action of U on X is induced by the conjugation of u on X. By Lemma 2.3 of [21] X has a proper ZU-submodule Y such that X/Y is a p-group. It is clear that Y is a finitely generated normal subgroup in G. Then by Theorem 21.3 of [13] F/Y = T/Y ×M/Y , where T/Y ∼ = C p ∞ and M/Y is some subgroup. If S = (M/Y )/τ(M/Y ) is not finitely generated then G is a nilpotent-by-Černikov group. Therefore S is a finitely generated p-divisible torsion-free group. This means that M/Y is a finite group. Hence F/Y 1 ∼ = Cp ∞ for some finitely generated G-invariant subgroup Y 1 , a contradiction. (2) Now suppose that the divisible part D is non-trivial. From the part (1) of this proof it follows that G/D is a polycyclic group or G is a nilpotent-by-Černikov group. Assume that G/D is a polycyclic quotient group. Then 〈F, u〉 is a finitely generated nilpotent-by-finite subgroup and moreover 〈F, u〉 = N ⋊ 〈u〉 for some abelian subgroup N of A. Lemma 3.2 yields that D ⋊ 〈u s 〉 is a nilpotent subgroup for some integer s. Furthermore, the subgroup N ⋊ 〈u m 〉 is nilpotent for some integer m. Put k = min{m, s}. Then A ⋊ 〈u k 〉 is a nilpotent subgroup of finite index in G, a contradiction. Proposition 3.4. Let G = NU be the product of a normal nilpotent subgroup N and a polycyclic abelian subgroup U. If G satisfies Max-NČ, then it is of one of the following types: (i) G is a polycyclic group; (ii) G is a nilpotent-by-Černikov group;
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