On the structure of groups admitting faithful modules with certain conditions of primitivity

. In the paper we study structure of soluble-by-(cid:28)nite groups of (cid:28)nite torsion-free rank which admit faithful modules with conditions of primitivity. In particular, we prove that under some additional conditions if an in(cid:28)nite (cid:28)nitely generated linear group G of (cid:28)nite rank admits a fully primitive fully faithful module then G has in(cid:28)nite FC -centre

Àíîòàöiÿ. Ó ñòàòòi äîñëiäaeó¹òüñÿ ñòðóêòóðà ìàéaeå ðîçâ9ÿçíèõ ãðóï ñêií÷åííîãî âiëüíîãî ðàíãóD ÿêi äîïóñêàþòü òî÷íi ìîäóëi ç óìîâàìè ïðèìiòèâíîñòiF ÇîêðåìàD äîâåäåíîD ùî çà äåÿêèõ äîäàòêîâèõ óìîâD ÿêùî íåñêií÷åííà ñêií÷åííî ïîðîäaeåíà ëiíiéíà ãðóïà G ñêií÷åííîE ãî ðàíãó äîïóñêà¹ öiëêîì ïðèìiòèâíèé öiëêîì òî÷íèé ìîäóëüD òî G ìà¹ íåñêií÷åííèé F CEöåíòðF Êëþ÷îâi ñëîâà: íiëüïîòåíòíi ãðóïèD ìiíiìàêñíi ãðóïèD ãðóïîâi êiëüE öÿD iíäóêîâàíi ìîäóëi MSC2020: Pri 16S34, Sec 20C07, 11R27 A group G is said to have nite (Prufer) rank if there is a positive integer m such that any nitely generated subgroup of G may be generated by m elements; the smallest m with this property is the rank r(G) of G. A group G is said to be of nite torsion-free rank if it has a nite series each of whose factor is either innite cyclic or locally nite, the number r 0 (G) of innite cyclic factors in such a series is the torsion-free rank of G. If a group G has a nite series each of whose factor is either cyclic or quasi-cyclic then G is said to be minimax, the number m(G) of innite factors in this series is the minimax length of the group G. If in such a series all factors are cyclic then the group G is said to be polycyclic. It is well known that any nitely generated nilpotent group is polycyclic.
Let H be a subgroup of a group G, the subgroup H is said to be dense in G if for any g ∈ G there is an integer such that g n ∈ H. If g n ∈ G\H for any and any g ∈ G\H then the subgroup H is said to be isolated in G. If the group G is locally nilpotent then the isolator is G (H) = {g ∈ G | g n ∈ H f or some n ∈ N} of H in G is a subgroup of G and if H is a normal subgroup then so is is G (H).
Let H be a subgroup of a group G and let U be a right RH-module. The tensor product U ⊗ RH RG is named the RG-module induced from the RH- Evidently, the structure of the group G has a decisive inuence on the properties of the RG-module M . However, if the group G admits a faithful RGmodule, then the properties of the module M may have signicant impacts on the structure of the group G. An important result of this type was obtained by Gaschutz in [3] where he proved that a nite group G admits a simple faithful kG-module, where k is a eld of characteristic zero, if and only if the socle Soc(G) may be generated by one class of conjugated elements, where Soc(G) is the subgroup generated by all minimal normal subgroups of G. In [7], this result was generalized on the class of locally normal groups. In [14] we studied soluble groups of nite rank and in [6] we considered some locally nite groups which admit faithful simple kG-modules. Let ϕ be a representation of a group G over a eld k and let M be an kG-module of the representation ϕ. The representation ϕ is said to be faithful if Kerϕ = 1. If M is induced from some F H-module U , where H is a subgroup of the group G, then we say that the representation ϕ is induced from a representation φ of the subgroup H, where U is the module of the representation φ. The module M and the representation ϕ are said to be primitive if there are no subgroups H < G such that M is induced from an F H-submodule.
In [11] we proved that if a minimax nilpotent group G of class 2 admit a faithful simple primitive kG-module, where k is a nitely generated eld of characteristic zero then the group G is polycyclic. In [10] we showed that in the class of soluble groups of nite rank with the maximal condition for normal subgroups only polycyclic groups may have faithful irreducible primitive representations over a eld of characteristic zero.
Let R be a ring and let M , X and Y be R-modules. The modules X and Y are separated in M if X and Y don't have nonzero isomorphic sections which are isomorphic to a submodule of M . The R-module M is said to be ρ-critical if K R (M ) = ρ and K R (M/U ) < K R (M ) = ρ for any non-zero submodule Let N be a nilpotent group of nite torsion-free rank, let k be a eld and let M be an kN -module. Let S be a nitely generated subring of k. Let 0 = a ∈ M and let H be a proper isolated normal subgroup of N . We say that (a, H) is an important pair for the kN -module M if there is a nitely generated dense subgroup A ≤ N such that: (i) the module aSA is ρ-critical and K SX (aSA) ≤ K SX (xSX) for any element 0 = x ∈ M and any nitely generated dense subgroup X ≤ N ; The module M is said to be impervious if it has no important pairs for any nitely generated subring S of k.
Theorem 1. (Cf. [15,Proposition 5.2]) Let N be a nilpotent non-abelian minimax torsion-free normal subgroup of a soluble-by-nite group G of nite torsion-free rank. Let K be a G-invariant subgroup of N such that the quotient group N/K is torsion-free abelian. Let k be a eld of characteristic zero. Let W be a kG-module which has kN -torsion and for any proper isolated Ginvariant Lemma 1. Let A be an abelian minimax group acted by a nitely generated group G. Suppose that A = T ×D, where T is a torsion group and D is a nitely generated torsion-free group. Then for any nitely generated dense subgroup B of A the group A has a nitely generated dense G-invariant subgroup C such that B ≤ C.
Proof. At rst, we should note that T is a union of an ascending series of its nite G-invariant subgroups. As the subgroup D is nitely generated, D = d 1 , ..., d t . Suppose that G = g 1 , ..., g m . As A = T × D, for any d i and g j there are u ij ∈ T and v ij ∈ D such that d i g j = u ij v ij , where 1 ≤ i ≤ t and 1 ≤ j ≤ m. Since T is an union of an ascending series of its nite G-invariant subgroups, there is a nite G-invariant subgroup U ≤ T such that all u ij ∈ U and therefore, as d i and 1 ≤ j ≤ m. It implies that U × D = C is a nitely generated dense G-invariant subgroup of A. As the subgroup B is nitely generated, we can choose the subgroup U ≤ T such that B ≤ U × D = C.
Lemma 2. Let N be a minimax nilpotent torsion-free group acted by a nitely generated group G. Suppose that the quotient group N/is N (N ) isnitely generated and the quotient group is N (N )/N is innite. Then for any UR nitely generated dense subgroup H of N the group N has a normal dense Ginvariant subgroup M such that H ≤ M , the quotient group N/M is torsion and m(M ) < m(N ).
Proof. As the quotient group A = N/N is minimax abelian, it is well known that A = T × D, where T = is N (N )/N and, as the the quotient group N/is N (N ) is nitely generated, the subgroup D is nitely generated. Then, by Lemma 1, there is a nitely generated dense G-invariant subgroup C = M/N such that B ≤ C, where B = HN /N , and hence H ≤ M . Since N ≤ M , we can conclude that M is a normal subgroup of N . As the quotient group T = is N (N )/N is innite torsion and the quotient group C = M/N is nitely generated, we can conclude that m(M ) < m(N ).
Lemma 3. Let N be a minimax nilpotent torsion-free group. Suppose that the quotient group N/is N (N ) is nitely generated. Let M be a normal subgroup of N such that the quotient group N/M is torsion abelian. Then the quotient group M/is M (M ) is nitely generated.
Proof. Suppose that the quotient group M/is N (M ) is not nitely generated.
Then the minimax nilpotent quotient group A = N/is N (M ) has an abelian torsion-free subgroup B = M/is M (M ) which is not nitely generated and such that the quotient group A/B is torsion abelian. Passing to the quotient group A/t(A) we can assume that the group A is torsion free. Then, as the quotient group A/B is torsion, it follows from [5, Theorem 17.3.1] that the group A is abelian. Thus, the group N has the torsion-free abelian quotient group A which is not nitely generated but this is impossible because the quotient group N/is N (N ) is nitely generated. Proposition 1. Let N be a minimax nilpotent torsion-free group acted by a nitely generated group G. Suppose that the quotient group N/is N (N ) is nitely generated. Then for any nitely generated dense subgroup H of N the group N has a nitely generated dense G-invariant subgroup S such that H ≤ S.
Proof. The proof is by induction on the minimax length m(N ) of the group N . Suppose that the quotient group is N (N )/N is nite then the quotient group N/N is nitely generated and hence, by [5,Theorem 16.2.5], so is the group N . Thus, the assertion is evident. So, we can assume that the quotient group is N (N )/N is innite. Then, by Lemma 2, there is a dense normal Ginvariant subgroup M of N such that H ≤ M , the quotient group N/M is torsion and m(M ) < m(N ). By Lemma 3, the quotient group M/is N (M ) is nitely generated and m(M ) < m(N ). Then we can apply the induction hypothesis to the subgroup M .
The following theorem species the result of Theorem 1.

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Theorem 2. Let N be a nilpotent non-abelian minimax torsion-free normal subgroup of a nitely generated soluble-by-nite group G of nite torsion-free rank. Let k be a eld of characteristic zero. Let W be a kG-module which has kN -torsion and for any proper isolated Ginvariant subgroup X of N the module W is kX-torsion-free. Suppose that W is impervious as a kN -module and Sep (G,Y ) (xkG, xkY ) = G for any element 0 = x ∈ W and any G-invariant subgroup Y of N . Then for any nitely generated dense subgroup H of N the group N has a nitely generated dense G-invariant subgroup S such that H ≤ S.
Proof. The proof is by induction on the minimax length m(N ) of the group N . We put K = is N (N ) in the denotations of Theorem 1. Then, by Theorem 1, there is a dense G-invariant subgroup D ≤ N such that K ≤ D and the quotient group D/K is polycyclic. Since the quotient group N/D is an union of an ascending series of its nite G-invariant subgroups and the subgroup H is nitely generated, we can choose the subgroup H such that H ≤ D. If m(D) < m(N ) then we can apply the induction hypothesis. So, we can assume that m(D) = m(N ) and hence that the quotient group N/D is nite. It implies that the quotient group N/is N (N ) is polycyclic and hence it is nitely generated. Then the assertion follows from Proposition 1, where the action of G on N is given by conjugations.
Let G be a group. We say that an RG-module W is fully primitive if any RG-submodule of W is not induced from any RH-submodule for any subgroup H < G.
Theorem 3. Let N be a nilpotent non-abelian torsion-free normal subgroup of a nitely generated linear group G of nite rank. Let k be a eld of characteristic zero and let W be a fully primitive kG-module. Suppose that W is kNtorsion and for any proper isolated Ginvariant subgroup X of N the module W is kX-torsion-free. Then for any nitely generated dense subgroup H of N the group N has a nitely generated dense G-invariant subgroup S such that H ≤ S.
Proof. It follows from [15, Lemma 6.1.] that the group G is minimax-bynite. It implies that G is a nitely generated soluble-by-nite group of nite torsion-free rank and the subgroup N is minimax. As the module W is strongly primitive, it follows from [10, Lemma 3.1.4(ii)] that Sep (G,Y ) (xkG, xkY ) = G for any element 0 = x ∈ W and any G-invariant subgroup Y of N . If the module W is not impervious as a kN -module then it follows from [10, Proposition 3.2.3] that there are an element 0 = x ∈ W and a proper isolated subgroup Y ≤ N such that the module W is not kY-torsion-free and Sep (G,N ) (xkG, xkN ) ≤ N G (Y ). Since Sep (G,Y ) (xkG, xkY ) = G, we see that G = N G (Y ) but it is impossible because by the choice of N the module W is kY-torsion-free for any proper isolated subgroup Y ≤ N. Thus, W is impervious as a kN-module.
Then the assertion follows from Theorem 2. Corollary 1. Let N be a nilpotent non-abelian torsion-free normal subgroup of a nitely generated linear group G of nite rank. Let k be a nitely generated eld of characteristic zero and let W be a fully primitive kG-module. Suppose that W has kN -torsion and for any proper isolated Ginvariant subgroup X of N the module W is kX-torsion-free. Then there are a G-invariant polycyclic dense subgroup S of N and an element 0 = a ∈ W such that akN = akS⊗ kS kN .
Proof. By [10, Lemma 3.2.2(ii)], there are an element 0 = a ∈ W and a nitely generated dense subgroup H of N such that akN = akH⊗ kH kN . Then, by Theorem 3, there is a G-invariant polycyclic dense subgroup S of N such that H ≤ S. As H ≤ S, we can conclude that akN = akS⊗ kS kN .
We say that a module over a group ring RG is fully faithful for G if every non-zero submodule has trivial centralizer in G. If G is a group then the F Ccenter ∆(G) = {g ∈ G| |G : C G (g)| < ∞}of G is a characteristic subgroup of G. By [15, Theorem 6.1], if an innite nitely generated linear group G of nite rank admits a faithful primitive irreducible kG-module, where k is a eld of characteristic zero, then |∆(G)| = ∞.
Theorem 4. Let G be a innite nitely generated linear group of nite rank. Let k be a eld of characteristic zero and let W be a fully primitive and fully faithful kG-module. Then either |∆(G)| = ∞ or there is a nilpotent minimax torsion-free normal subgroup Lof G such that the quotient group G/Lis polycyclic-by-nite and the module W is kL-torsion-free.
Proof. By lemma 6.1, G has a nite series L ≤ G 0 ≤ G of normal subgroups such that |G/G 0 | <∞, the quotient group G 0 /L is polycyclic and the subgroup L is torsion-free nilpotent minimax which has no non-abelian torsion-free polycyclic G-sections. Evidently, the group G is polycyclic-by-nite if and only if the subgroup L is trivial. If the group G is polycyclic-by-nite then the assertion follows from [4, Theorem A]. So, we can assume that the subgroup L is not trivial. Suppose that |∆(G)| < ∞ and the module W is not kL-torsionfree. Then there is a G-invariant subgroup N ≤ L such that W has kN -torsion and for any proper isolated Ginvariant subgroup X of N the module W is kX-torsion-free. If the subgroup N is non-abelian then, by Theorem 3, it has a dense nitely generated G-invariant subgroup but this is impossible because N ≤ L and L has no non-abelian torsion-free polycyclic G-sections. So, the subgroup N is abelian. As the subgroup N is torsion-free and |∆(G)| < ∞, we can conclude that ∆(G) ∩ N = ∆ G (N) = 1. Then, as the module W is fully primitive, it follows from [8, Theorem 4.2] that there is an element 0 = a ∈ W such that r 0 (G/C G (akG)) < r 0 (G). It implies that C G (akG) = 1 but it is impossible because the module W is fully faithful. The obtained contradiction shows that if |∆(G)| < ∞ then the module W is kL-torsion-free.