On homotopy nilpotency

We review established and recent results on the homotopy nilpotence of spaces. In particular, the homotopy nilpotency of the loop spaces \(\Omega(G/K)\) of homogenous spaces \(G/K\) for a compact Lie group \(G\) and its closed homotopy nilpotent subgroup \(K \lt G\) is discussed.

Compact groups with a set of positive Haar measure satisfying a nilpotent law

The following question is proposed by Martino, Tointon, Valiunas and Ventura in [4, question 1·20]: Let G be a compact group, and suppose that \[\mathcal{N}_k(G) = \{(x_1,\dots,x_{k+1}) \in G^{k+1} \;|\; [x_1,\dots, x_{k+1}] = 1\}\] has positive Haar measure in $G^{k+1}$ . Does G have an open k-step nilpotent subgroup? We give a positive answer for $k = 2$ .

On nilpotent automorphism groups of function fields

We study the automorphisms of a function field of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. More precisely, we show that the order of a nilpotent subgroup G of its automorphism group is bounded by 16 (g – 1) when G is not a p-group. We show that if |G| = 16(g – 1), then g – 1 is a power of 2. Furthermore, we provide an infinite family of function fields attaining the bound.

ON THE RANK OF A VERBAL SUBGROUP OF A FINITE GROUP

We show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

A right Engel sink of an element g of a group G is a set $${{\mathscr {R}}}(g)$$ R ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[g,x],x],\dots ,x]$$ [ . . . [ [ g , x ] , x ] , ⋯ , x ] belong to $${\mathscr {R}}(g)$$ R ( g ) . (Thus, g is a right Engel element precisely when we can choose $${{\mathscr {R}}}(g)=\{ 1\}$$ R ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set $${{\mathscr {E}}}(g)$$ E ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[x,g],g],\dots ,g]$$ [ . . . [ [ x , g ] , g ] , ⋯ , g ] belong to $${{\mathscr {E}}}(g)$$ E ( g ) . (Thus, g is a left Engel element precisely when we can choose $${\mathscr {E}}(g)=\{ 1\}$$ E ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.

On semi-σ-nilpotent finite groups

Let [Formula: see text] be a partition of the set [Formula: see text] of all primes and [Formula: see text] a finite group. A chief factor [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-central if the semidirect product [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. [Formula: see text] is called [Formula: see text]-nilpotent if every chief factor of [Formula: see text] is [Formula: see text]-central. We say that [Formula: see text] is semi-[Formula: see text]-nilpotent (respectively, weakly semi-[Formula: see text]-nilpotent) if the normalizer [Formula: see text] of every non-normal (respectively, every non-subnormal) [Formula: see text]-nilpotent subgroup [Formula: see text] of [Formula: see text] is [Formula: see text]-nilpotent. In this paper we determine the structure of finite semi-[Formula: see text]-nilpotent and weakly semi-[Formula: see text]-nilpotent groups.

Low-complexity computations for nilpotent subgroup problems

We solve the following algorithmic problems using [Formula: see text] circuits, or in logspace and quasilinear time, uniformly in the class of nilpotent groups with bounded nilpotency class and rank: subgroup conjugacy, computing the normalizer and isolator of a subgroup, coset intersection, and computing the torsion subgroup. Additionally, if any input words are provided in compressed form as straight-line programs or in Mal’cev coordinates, the algorithms run in quartic time.

The largest conjugacy class size and the nilpotent subgroups of finite groups

LetHbe a nilpotent subgroup of a finite nonabelian groupG, let{\pi=\pi(|H|)}and let{{\operatorname{bcl}}(G)}be the largest conjugacy class size of the groupG. In the present paper, we show that{|HO_{\pi}(G)/O_{\pi}(G)|<{\operatorname{bcl}}(G)}.

A NILPOTENCY-LIKE CONDITION FOR INFINITE GROUPS

If $k$ is a positive integer, a group $G$ is said to have the $FE_{k}$-property if for each element $g$ of $G$ there exists a normal subgroup of finite index $X(g)$ such that the subgroup $\langle g,x\rangle$ is nilpotent of class at most $k$ for all $x\in X(g)$. Thus, $FE_{1}$-groups are precisely those groups with finite conjugacy classes ($FC$-groups) and the aim of this paper is to extend properties of $FC$-groups to the case of groups with the $FE_{k}$-property for $k>1$. The class of $FE_{k}$-groups contains the relevant subclass $FE_{k}^{\ast }$, consisting of all groups $G$ for which to every element $g$ there corresponds a normal subgroup of finite index $Y(g)$ such that $\langle g,U\rangle$ is nilpotent of class at most $k$, whenever $U$ is a nilpotent subgroup of class at most $k$ of $Y(g)$.