Related papers: Nilpotent $p$-local finite groups

For a finite group $G$ and an element $x\in G$, the subset $$ nil_G(x)=\{y\in G \mid <x,y>~~ is ~~ nilpotent\}$$ is called nilpotentizer of $x$ in $G$. In this paper, we give two solvabilty criteria for a finite group by the structure and…

Let $G$ be the generalized free product of two groups with an amalgamated subgroup. We propose an approach that allows one to use results on the residual $p$-finiteness of $G$ for proving that this generalized free product is residually a…

Let $A$ be a non-metacyclic finite group. Suppose that $A$ acts coprimely on a finite group $G$ in such a manner that $C_G(a)$ is nilpotent for any $a\in A^{\#}$. In the present paper we investigate some conditions on $A$ which imply that…

The main goal of the paper is to present a general model theoretic framework to understand a result of Shalev on probabilistically finite nilpotent groups. We prove that a suitable group where the equation $[x_1,\ldots,x_k]=1$ holds on a…

We give effective proofs of residual finiteness and conjugacy separability for finitely generated nilpotent groups. In particular, we give precise asymptotic bounds for a function introduced by Bou-Rabee that measures how large the…

For a finite group $G$, let $LC(G)$ be the subgroup generated by elements $x$ such that, for all $y \in G$ and all integers $n$, the order of $x^n y$ divides the least common multiple of the orders of $x$ and $y$. This subgroup is a…

We extend the characterization of abelian groups with ramification structures given by Garion and Penegini to finite nilpotent groups whose Sylow $p$-subgroups have a `nice power structure', including regular $p$-groups, powerful $p$-groups…

A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains…

Let $\mathcal F$ be a saturated fusion system on a finite $p$-group $S$, and let $P$ be a strongly $\mathcal F$-closed subgroup of $S$. We define the concept ``$\mathcal F$-essential subgroups with respect to $P$" which are some proper…

Let $FH$ be a supersolvable Frobenius group with kernel $F$ and complement $H$. Suppose that a finite group $G$ admits $FH$ as a group of automorphisms in such a manner that $C_G(F)=1$ and $C_{G}(H)$ is nilpotent of class $c$. We show that…

Suppose that a metacyclic Frobenius group $FH$, with kernel $F$ and complement $H$, acts by automorphisms on a finite group $G$, in such a way that $C_G(F)$ is trivial and $C_G(H)$ is nilpotent. It is known that $G$ is nilpotent and its…

We consider ordered tuples in finite groups generating nilpotent subgroups. Given an integer $q$ we consider the poset of nilpotent subgroups of class less than $q$ and its corresponding coset poset. These posets give rise to a family of…

In this paper we find a necessary and sufficient condition for a finite nilpotent group to have an abelian central automorphism group.

Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset Delta of a finite group G is called a p-base (where p is a prime) if…

We give a new, unexpected characterization of saturated fusion systems on a p-group S in terms of idempotents in the p-local double Burnside ring of S that satisfy a Frobenius reciprocity relation, and reformulate fusion-theoretic phenomena…

We prove that a finitely generated Lie algebra $L$ such that (i) every commutator in generators is ad-nilpotent, and (ii) $ L$ satisfies a polynomial identity, is nilpotent. As a corollary we get that a finitely generated residually-$p$…

A finite group $G$ is called {\it $p^i$-central of height $k$} if every element of order $p^i$ of $G$ is contained in the $k^{th}$-term $\zeta_k(G)$ of the ascending central series of $G$. If $p$ is odd such a group has to be $p$-nilpotent…

We investigate some situation in which automorphisms of a group G are uniquely determined by their restrictions to a proper subgroup H. Much of the paper is devoted to studying under which additional hypotheses this property forces G to be…

Let P be a finite metacyclic 2-group and F a fusion system on P. We prove that F is nilpotent unless P has maximal class or P is homocyclic, i.e. P is a direct product of two isomorphic cyclic groups. As a consequence we obtain the…

Let $G$ be a finite group with the property that if $a,b$ are powers of $\delta_1^*$-commutators such that $(|a|,|b|)=1$, then $|ab|=|a||b|$. We show that $\gamma_{\infty}(G)$ is nilpotent.