### $\U0001d51bC$-elements in groups and Dietzmann classes.

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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 $G/{Z}_{n}\left(G\right)$ is finite. For $k<4$, such $n$ depends only on $k$.

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...

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic...

Let G be a non-periodic locally solvable group. A characterization is given of the subgroups-D of G for which the map $X\to X\cap D$, for all $X\le G$, defines a lattice-endomorphism.