Fitting quotients of finitely presented abelian-by-nilpotent groups (original) (raw)

A sufficient condition for finite presentability of abelian-by-nilpotent groups

Groups, Geometry, and Dynamics, 2014

A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a geometric procedure that generalizes the construction of finitely presented metabelian groups introduced by R. Mathematics Subject Classification (2010) 20F16, 20F65. in a series of papers in the late 1990s: in [5], Brookes, Roseblade and Wilson showed that a finitely presented abelian-by-polycyclic group G is necessarily nilpotent-by-nilpotent-byfinite; in then, this conclusion is sharpened to G is nilpotent, by nilpotent of class at most two, by finite.

Class two nilpotent capable groups

Bulletin of the Australian Mathematical Society, 1996

We give a bound for the number of generators for groups of exponent p depending on the rank of the centre when centre and commutator subgroup coincide, provided that the group is isomorphic to the quotient group modulo the centre of some group.

The congruence subgroup problem for finitely generated nilpotent groups

Journal of Group Theory

The congruence subgroup problem for a finitely generated group Γ and for G ≤ Aut ⁢ ( Γ ) G\leq\mathrm{Aut}(\Gamma) asks whether the map G ^ → Aut ⁢ ( Γ ^ ) \hat{G}\to\mathrm{Aut}(\hat{\Gamma}) is injective, or more generally, what its kernel C ⁢ ( G , Γ ) C(G,\Gamma) is. Here X ^ \hat{X} denotes the profinite completion of 𝑋. In the case G = Aut ⁢ ( Γ ) G=\mathrm{Aut}(\Gamma) , we write C ⁢ ( Γ ) = C ⁢ ( Aut ⁢ ( Γ ) , Γ ) C(\Gamma)=C(\mathrm{Aut}(\Gamma),\Gamma) . Let Γ be a finitely generated group, Γ ¯ = Γ / [ Γ , Γ ] \bar{\Gamma}=\Gamma/[\Gamma,\Gamma] , and Γ * = Γ ¯ / tor ⁢ ( Γ ¯ ) ≅ Z ( d ) \Gamma^{*}=\bar{\Gamma}/\mathrm{tor}(\bar{\Gamma})\cong\mathbb{Z}^{(d)} . Define Aut * ⁢ ( Γ ) = Im ⁡ ( Aut ⁢ ( Γ ) → Aut ⁢ ( Γ * ) ) ≤ GL d ⁢ ( Z ) . \mathrm{Aut}^{*}(\Gamma)=\operatorname{Im}(\mathrm{Aut}(\Gamma)\to\mathrm{Aut}(\Gamma^{*}))\leq\mathrm{GL}_{d}(\mathbb{Z}). In this paper we show that, when Γ is nilpotent, there is a canonical isomorphism C ⁢ ( Γ ) ≃ C ⁢ ( Aut * ⁢ ( Γ ) , Γ ...

On a Class of Generalized Nilpotent Groups

Journal of Algebra, 2002

We explore the class of generalized nilpotent groups in the universe c of all radical locally finite groups satisfying min-p for every prime p. We obtain that this class is the natural generalization of the class of finite nilpotent groups from the finite universe to the universe c. Moreover, the structure of-groups is determined explicitly. It is also shown that is a subgroup-closed c-formation and that in every c-group the Fitting subgroup is the unique maximal normal-subgroup.

On Torsion-by-Nilpotent Groups

Journal of Algebra, 2001

Let C C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C C are torsion-by-nilpotent, then all soluble groups of C C are torsion-by-nilpotent. From that, we deduce the following conse-Ž quence, similar to a well-known result of P. Hall 1958, Illinois J. Math. 2,. 787᎐801 : if H is a normal subgroup of a group G such that H and GrHЈ are Ž. Ž. locally finite-by-nilpotent, then G is locally finite-by-nilpotent. We give an Ž. example showing that this last statement is false when '' locally finite-by-nilpotent'' is replaced with ''torsion-by-nilpotent.''

On the Capability of Finitely Generated Non-Torsion Groups of Nilpotency Class 2

Glasgow Mathematical Journal, 2011

A group is called capable if it is a central factor group. In this paper, we establish a necessary condition for a finitely generated non-torsion group of nilpotency class 2 to be capable. Using the classification of two-generator non-torsion groups of nilpotency class 2, we determine which of them are capable and which are not and give a necessary and sufficient condition for a two-generator non-torsion group of class 2 to be capable in terms of the torsion-free rank of its factor commutator group.

On finite products of nilpotent groups

Archiv der Mathematik, 1994

i. Introduetion. A well-known theorem of Kegel [7] and Wielandt [9] states the solubility of every finite group G = AB which is the product of two nilpotent subgroups A and B; see [1], Theorem 2.4.3. In order to determine the structure of these groups it is of interest to know which subgroups of G are conjugate (or at least isomorphic) to a subgroup that inherits the factorization. A subgroup S of the factorized group G = AB is called prefactorized if S = (A c~ S) (B ~ S), it is called factorized if, in addition, S contains the intersection A c~ B. Generally, even characteristic subgroups of G are not prefactorized, as can be seen e.g. from Examples 1 and 2 below.

The Nilpotency Class of Fitting Subgroups of Groups with Basis Property

2012

A group G is called group with basis property if for any subgroup H of G there exists a basis, i.e. a minimal generating set such that any two bases of H are equivalent. We show that any group with basis property is Frobenies group with kernel p-group P = F it(G) and complement q-group < y > of order q b. Let G be group with basis property, then F it(G) in general has no upper bound, but in some special cases the nilpotency class of F it(G) is bounded and that depends on the order of the complement group. We proved the following theorem: Let G be a finite group with basis property which is not a p−group, then the nilpotency class of the Fitting subgroup F it(G) has no upper bound.Moreover, for each prime number q > 2there exists a nite group with the basis property such that: c(F it(G) = q − 1; | < y > | = q.