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A nilpotency condition for finitely generated soluble groups

Costantino Delizia (1998)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove that if k > 1 is an integer and G is a finitely generated soluble group such that every infinite set of elements of G contains a pair which generates a nilpotent subgroup of class at most k , then G is an extension of a finite group by a torsion-free k -Engel group. As a corollary, there exists an integer n , depending only on k and the derived length of G , such that G / Z n G is finite. For k < 4 , such n depends only on k .

A note on supersoluble maximal subgroup and theta-pairs.

James C. Beidleman, Howard Smith (1993)

Publicacions Matemàtiques

A θ-pair for a maximal subgroup M of a group G is a pair (A, B) of subgroups such that B is a maximal G-invariant subgroup of A with B but not A contained in M. θ-pairs are considered here in some groups having supersoluble maximal subgroups.

Almost fixed-point-free automorphisms of prime order

Bertram Wehrfritz (2011)

Open Mathematics

Let ϕ be an automorphism of prime order p of the group G with C G(ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are...

Asymptotic dimension of discrete groups

A. Dranishnikov, J. Smith (2006)

Fundamenta Mathematicae

We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [B-D2] to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their Hirsch length.

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