Related papers: Lower central words in finite $p$-groups

A first-order formula is called primitive positive (pp) if it only admits the use of existential quantifiers and conjunction. Pp-formulas are a central concept in (fixed-template) constraint satisfaction since CSP($\Gamma$) can be viewed as…

Let G be a finite group with Sylow p-subgroup P. We show that the character table of G determines whether P has maximal nilpotency class and whether P is a minimal non-abelian group. The latter result is obtained from a precise…

We consider word automaticity for groups that are nilpotent of class $2$ and have exponent a prime $p$. We show that the infinitely generated free group in this variety is not word automatic. In contrast, the infinite extra-special…

Given a group-word w and a group G, the verbal subgroup w(G) is the one generated by all w-values in G. The word w is called concise if w(G) is finite whenever the set of w-values in G is finite. It is an open question whether every word is…

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p; then G contains a simple group as a subquotient which exhibits the same property. In addition…

We obtain certain results on a finite $p$-group whose central automorphisms are all class preserving. In particular, we prove that if $G$ is a finite $p$-group whose central automorphisms are all class preserving, then $d(G)$ is even, where…

Using methods of associative algebras, Lie theory, group cohomology, and modular representation theory, we construct profinite $p$-adic analytic groups such that the centralizer of each of their non-trivial elements is abelian. The paper…

Let $\mathcal C$ be a set of finite groups which is closed under taking subgroups and let $d$ and $M$ be positive integers. Suppose that for any $G\in\mathcal C$ whose order is divisible by at most two distinct primes there exists an…

For a set $A$ of $k$ elements from an additive abelian group $G$ and a positive integer $r \leq k$, we consider the set of elements of $G$ that can be written as a sum of $h$ elements of $A$ with at least $r$ distinct elements. We denote…

If $G$ is a finite Abelian group, define $s_{k}(G)$ to be the minimal $m$ such that a sequence of $m$ elements in $G$ always contains a $k$-element subsequence which sums to zero. Recently Bitz et al. proved that if $n = exp(G)$, then…

For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we…

It was shown in [A. Azimifard, E. Samei, N. Spronk, JFA 2009; arxiv 0805.3685] that the ZL-amenability constant of a finite group is always at least 1, with equality if and only if the group is abelian. It was also shown in the same paper…

We prove the following results. Let w be a multilinear commutator word. If G is a profinite group in which all w-values are contained in a union of countably many periodic subgroups, then the verbal subgroup w(G) is locally finite. If G is…

Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results…

Let $A$ be an elementary abelian $r$-group with rank at least $3$ that acts faithfully on the finite $r'$-group $G$. Assume that $G$ is $A$-simple, so that $G = K_{1} \times\cdots\times K_{n}$ where $K_{1},\ldots,K_{n}$ is a collection of…

Let $G$ be a connected algebraic semisimple real Lie group with finite center and no compact factors, and let $\Gamma$ be a Zariski dense discrete subgroup of $G$. We show that $\Gamma$ contains free, finitely generated subsemigroups whose…

The Chermak-Delgado lattice of a finite group $G$ is a self-dual sublattice of the subgroup lattice of $G$. In this paper, we focus on finite groups whose Chermak-Delgado lattice is a subgroup lattice of an elementary abelian $p$-group. We…

Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $\delta_k$-values $a,b\in G$ of coprime orders. In the course of the…

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p > 0$. We consider connected reductive subgroups $X$ of $G$ that contain a given distinguished unipotent element $u$ of $G$. A result of…

We prove that if $L=\mbox{}^2F_4(2^{2n+1})'$ and $x$ is a nonidentity automorphism of $L$ then $G=\langle L,x\rangle$ has four elements conjugate to $x$ that generate $G$. This result is used to study the following conjecture about the…