‎In the classical group theory there is‎ an open question‎: ‎Is every torsion free n-Engel group (for n ≥ 4)‎, nilpotent?‎. ‎To answer the question‎, ‎Traustason‎ [11] showed that with some additional conditions all‎ ‎4-Engel groups are locally nilpotent‎. ‎Here‎, ‎we gave some partial‎ answer to this question on Engel fuzzy subgroups‎. ‎We show that if μ is a normal 4-Engel fuzzy‎ subgroup of ...

a result of dixon‎, ‎evans and smith shows that if $g$ is a locally (soluble-by-finite) group whose proper subgroups are (finite rank)-by-abelian‎, ‎then $g$ itself has this property‎, ‎i.e‎. ‎the commutator subgroup of $g$ has finite rank‎. ‎it is proved here that if $g$ is a locally (soluble-by-finite) group whose proper subgroups have minimax commutator subgroup‎, ‎then also the commutator s...

we describe under various conditions abelian subgroups of the automorphism‎ ‎group $mathrm{aut}(t_{n})$ of the regular $n$-ary tree $t_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $tau =(e‎, ‎dots‎, ‎e,tau )sigma _{tau‎ ‎}$ where $sigma _{tau }$ is the $n$-cycle $left( 0,1‎, ‎dots‎, ‎n-1right) $‎. ‎as‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that...

In the present note, we discuss certain observations made by the author in February 2009 concerning strongly torsion-free profinite groups [cf. [Mzk2], Definition 1.1, (iii)]. These observations grew out of e-mail correspondences between the author, Akio Tamagawa, and Marco Boggi, as well as oral discussions between the author and Akio Tamagawa. Definition 1. Let G be a profinite group. (i) We ...

We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n p2, the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelia...

We give new examples of FA presentable torsion-free abelian groups. Namely, for every n > 2, we construct a rank n indecomposable torsion-free abelian group which has an FA presentation. We also construct an FA presentation of the group (Z,+) in which every nontrivial cyclic subgroup is not FA recognizable.

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsion-free subgroup N ; the quotient G/N always has maximal definably compact ...

We present about twenty conjectures, problems and questions about flat manifolds. Many of them build the bridges between the flat world and representation theory of the finite groups, hyperbolic geometry and dynamical systems. We shall present here some conjectures, problems and open questions related to flat manifolds. By flat manifold we understand a compact closed (without boundary) Riemanni...

We show that if G is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group G∗, such that G is a subgroup of G∗ but G is not quasiconvex in G∗ . We also prove that any non-elementary subgroup of a torsion-free word hyperbolic group G contains a free group of rank two which is malnormal and quasiconvex in G.