A group word w is said to be strongly concise in a class C $\mathcal C$ of profinite groups if, for any G , either takes at least continuum many values or the verbal subgroup ( ) $w(G)$ finite. It conjectured that all words are groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture multilinear commutator words, as well some other particular words. They also every nilpotent group...

The aim of this paper is to prove Theorem 1.1 below, a generalization to virtually nilpotent spaces of a result of Wilkerson [13] and Sullivan [12]. We recall that a CW complex Y is virtually nilpotent if (i) Y is connected, (ii) 7r~ Y is virtually nilpotent (i.e. has a nilpotent subgroup of finite index) and (iii) for every integer n > 1, zr~Y has a subgroup of finite index which acts nilpoten...

Let H∗(Be) be the cohomology of the Springer fibre for the nilpotent element e in a simple Lie algebra g, on which the Weyl group W acts by the Springer representation. Let ΛV denote the ith exterior power of the reflection representation of W . We determine the degrees in which ΛV occurs in the graded representation H∗(Be), under the assumption that e is regular in a Levi subalgebra and satisf...

Among other things, we prove that the group of automorphisms fixing every normal subgroup of a (nilpotent of class c)-by-abelian group is (nilpotent of class ≤ c)-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian group is soluble of derived length at most 3. An example shows that this bound cannot be improved.

We prove that every ω-categorical, generically stable group is nilpotent-byfinite and that every ω-categorical, generically stable ring is nilpotent-by-finite.

This is a toy example to illustrate some of the techniques of fusion and transfer as described in Gorenstein’s text Finite Groups. Finite groups with a unique subgroup of order p (p an odd prime) are classified in terms of a cyclic p′-extension of relatively simple combinatorial data. This note is intended to address a slightly misstated exercise in somewhat more detail and to be a toy example ...

for any group g, let c(g) denote the set of centralizers of g.we say that a group g has n centralizers (g is a cn-group) if |c(g)| = n.in this note, we prove that every finite cn-group with n ≤ 21 is soluble andthis estimate is sharp. moreover, we prove that every finite cn-group with|g| < 30n+1519 is non-nilpotent soluble. this result gives a partial answer to aconjecture raised by a. ashrafi in ...

We show that if the lower central series of the fundamental group of a closed oriented 3-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion 2-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental groups of closed oriented 3-manifolds.

Dyer and Formanek (1976) proved that if N is a free nilpotent group of class two and of rank 6= 1, 3, then the automorphism group Aut(N) of N is complete. The main result of this paper states that the automorphism group of an infinitely generated free nilpotent group of class two is also complete.

Mal’cev showed in the 1950s that there is a correspondence between radicable torsion-free nilpotent groups and rational nilpotent Lie algebras. In this paper we show how to establish the connection between the radicable hull of a finitely generated torsion-free nilpotent group and its corresponding Lie algebra algorithmically. We apply it to fast multiplication of elements of polycyclically pre...