Soluble Groups Of Finite Rank Research Articles

Automorphisms of Finite Order of Soluble Groups of Finite Rank

We study the effect on sections of a soluble-by-finite group G of finite rank of an almost fixed-point-free automorphism φ of G of finite order. We also elucidate the structure of G if φ has order 4 and if G is also (torsion-free)-by-finite. The latter extends recent work of Xu, Zhou and Liu.

To the question of existence of exact irreducible representations of soluble groups of finite rank over locally finite field

We find characteristic subgroup of soluble torsion-free group of finite rank, whose structure determines sufficient conditions of existence of exact irreducible representations of the group over locally finite field.

Finite Homomorphic Images of Groups of Finite Rank

Let π be a finite set of primes. We prove that each soluble group of finite rank contains a finite index subgroup whose every finite homomorphic π-image is nilpotent. A similar assertion is proved for a finitely generated group of finite rank. These statements are obtained as a consequence of the following result of the article: Each soluble pro-π-group of finite rank has an open normal pronilpotent subgroup.

Remarks on Azarov’s work on soluble groups of finite rank

We present proofs of D. N. Azarov’s recent three theorems determining precisely when a soluble group of finite rank is residually a finite \(\pi \)-group for a specified finite set \(\pi \) of primes. Our proofs seem to be substantially shorter; they also apply to groups with a somewhat weaker notion of finite rank.

On the Residual Finiteness of Some Generalized Products of Soluble Groups of Finite Rank

Let G be a free product of residually finite virtually soluble groups A and B of finite rank with an amalgamated subgroup H, H 6= A and H 6= B. And let H contains a subgroup W of finite index which is normal in both A and B. We prove that the group G is residually finite if and only if the subgroup H is finitely separable in A and B. Also we prove that if all subgroups of A and B are finitely separable in A and B, respectively, all finitely generated subgroups of G are finitely separable in G.

Approximability of finite rank soluble groups by certain classes of finite groups

For soluble groups of finite rank we obtain the necessary and sufficient condition to be a virtually residually finite p-group. We also prove that a soluble group G of finite rank is residually π-finite for some finite set π of primes if and only if it has no subgroups of type Q and the torsion radical of G is a finite group.

Chains of idempotents in endomorphism monoids

If G is a group with finite Hirsch number and with its maximal locally finite normal subgroup satisfying the minimal condition on subgroups, e.g. if G is a finite extension of a torsion-free soluble group of finite rank, then there exists an integer k=k(G) such that for every subgroup H of G any chain of idempotents in the endomorphism monoid \(\operatorname{End}(H)\) of H has length at most k.

Hopficity and Co-Hopficity in Soluble Groups

We show that a soluble group satisfying the minimal condition for its normal subgroups is co-hopfian and that a torsion-free finitely generated soluble group of finite rank is hopfian. The latter property is a consequence of a stronger result: in a minimax soluble group, the kernel of an endomorphism is finite if and only if its image is of finite index in the group.

On primitive representations of soluble groups of finite rank

In the paper it is proved, in particular, that a group is polycyclic if and only if it is soluble of finite rank, satisfies the ascending chain condition for normal subgroups and admits a faithful irreducible primitive representation over a field of characteristic zero. Methods are developed that enable one to study induced representations of nilpotent and soluble groups of finite rank.

On the primitivity of group algebras of certain classes of soluble groups of finite rank

In this paper it is proved in detail that the group algebra of a finitely generated metabelian group of finite rank over a field of characteristic zero is primitive if and only if its -centre is trivial.

On soluble groups with finite homomorphic images of bounded rank

Let G be a group. We denote by 9r(G) the set of all finite homomorphic images of G. The study of the connection between the properties of G and of groups of the set ~'(G) is a traditional problem of group theory. In this connection the following question, proposed by Wilson (see [1, Question 12.90]), is of interest: is a finitely generated residually finite group G minimax if there exists a finitely generated soluble minimax group H such that 9V(G) = 9V(H)? In the present paper we show that G is also minimax if ~-(G) C_ ~'(H). We recall that a group is minimax if it has a finite subnormal series each factor of which satisfies either the minimality condition or the maximality condition for subgroups. The basic properties of minimax groups are discussed in [2]. Lemma 1. Let G be a residually finite group, and let H be a group of finite rank. If ~(G) C ~(H), then G does not contain infinite elementary Abelian p-subgroups. Proof. Suppose that G contains an infinite elementary Abelian p-subgroup A. Then G contains a finite elementary Abelian p-subgroup B such that [B I > pr, where r is the rank of H. Since G is a residually finite group, we see that G contains a normal subgroup R of finite index such that B fq R = 1. Then G = G/R contains an elementary Abelian p-subgroup/~ such that IBI > p~. Since G E 37(G) and ~-(G) C ~-(g), we have G E 5r(g). Therefore, r(G) pL The lemma has been proved. Lemma 2. Let G be a subgroup of the Cartesian product G = :~iGi of finite soluble groups Gi of bounded orders. Suppose that H is a group of finite rank. If ~(G) C 3U(H), then G is a finite group. Proof. Since the orders of Gi are bounded, it follows that the solubility lengths of Gi are also bounded and, hence, G is soluble. Suppose that G is an infinite group and show by induction on the solubility length of G that the latter contains an infinite elementary Abelian p-subgroup. Assume that G is an Abelian group. Since there exists n E N such that IGil < n for all i, it follows that there exists t E N such that G~ = 1 for all i; therefore, G t = 1. From this it follows that G contains an infinite elementary Abelian p-subgroup. Let D be a derived subgroup of G. If ]D I = :xD, then by the inductive hypothesis G contains an infinite elementary Abelian p-subgroup. Thus, one can assume that IDI < :xD. Since G is residually finite, there exists a normal subgroup R < G of finite index such that RND = 1 and, consequently, R is an infinite Abelian subgroup of G. As was shown above, R contains an infinite elementary Abelian p-subgroup. Thus, G contains an infinite elementary Abelian p-subgroup. This contradicts Lemma 1. Therefore, the group G is finite. The lemma has been proved. Lemma 3. Let G be a residually finite group, and let H be an almost metanilpotent soluble group of finite rank. If ~(G) C F(H), then G is an almost metanilpotent group. Proof. Since G is a residually finite group, we see that G is a subgroup of the Cartesian product = ~iGi, where Gi E JZ(G). The group H has the normal series

Some soluble groups of finite rank and some related matrix groups

Let G be a torsion-free soluble group of finite rank and F any field. The group algebra FG is an Ore domain; let D denote its division ring of quotients. It seems likely that D is always locally residually finite-dimensional over F. This is certainly so in the non-modular case. Here in some special situations we settle the modular case. We include some applications to groups of matrices.

Groups which satisfy a weak form of Poincaré duality

Our main result is that a “restricted Poincaré duality” property with respect to finite dimensional coefficient modules over a field holds for a certain class of groups which includes all soluble groups of finite Hirsch length. This relies on a generalisation to the given class of a module construction by Stammbach; an extension of his result on homological dimension to these groups is given. We also generalise the well-known result that torsion-free soluble groups of finite rank are countable.

On rings of quotients of group algebras of soluble groups of finite rank

Let F be a field and G a finite extension of a torsion-free soluble group of finite rank such that the characteristics of F does not lie in the spectrum of G. Thr group algebra FG has a classical ring D of quotients. We prove that D is locally residually finite-dimensional over F. There are a number of corollaries. Our results hold more generally. For example, G could also be any soluble-by-finite linear group over the rationals or any metabelian group with torsion-free derived subgroup.

On trifactorized soluble groups of finite rank

Let the soluble-by-finite group G=AB=AC=BC be the product of two nilpotent subgroups A and B and a subgroup C. It is shown that, if G has finite abelian section rank and C is hypercentral (hypercyclic), then G is hypercentral (hypercyclic). Moreover, if G is an L 1-group and C is nilpotent, then G is nilpotent.

The Cohomology of Soluble Groups of Finite Rank

The Cohomology of Soluble Groups of Finite Rank

Decision problems for soluble groups of finite rank

Decision problems for soluble groups of finite rank

Normal series of soluble groups of finite rank

Normal series of soluble groups of finite rank

On products of soluble groups of finite rank

On products of soluble groups of finite rank

Ideals in group rings of soluble groups of finite rank

The original motivation for this paper was the question of primitivity for group rings of soluble groups of finite rank. At the end this is touched upon as an application of a theorem about prime ideals in such rings. If a group Γ acts on a set S we say an element is (Γ)-orbital if its orbit is finite and write ΔΓ(S) for the subset of such elements. The FC-radical of a group G, denoted by Δ(G), is just ΔG(G) where the action of G on itself is by conjugation.