Related papers: Quasirandom groups

For a finite abelian group $(G,+)$, the constant $C(G)$ is defined to be the smallest natural number $k$ such that any sequence in $G$ having length $k$ will have a subsequence of consecutive terms whose sum is zero. For a subset…

We produce a simple group $G$ of cardinality $\aleph_1$ which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset $Y \subseteq G$ there exists a natural…

Given a group $G = H_1 \ast_A H_2$ which is the free product of two finitely generated groups $H_1$ and $H_2$ with amalgamation over a cyclic subgroup $A$ which is malnormal in $G$, we study relations between the structure of its subgroups…

The free semigroup $\mathcal{F}$ over a finite alphabet $\mathcal{A}$ is the set of all finite words with letters from $\mathcal{A}$ equipped with the operation of concatenation. A subset $S$ of $\mathcal{F}$ is $k$-product-free if no…

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is…

Let $G$ be a group, and $S$ a non-empty subset of $G$. Then $S$ is \emph{product-free} if $ab\notin S$ for all $a, b \in S$. We say $S$ is \emph{locally maximal product-free} if $S$ is product-free and not properly contained in any other…

We show that a finite zero-sum-free sequence $\alpha$ over an abelian group has at least $c|\alpha|^{4/3}$ distinct subsequence sums, unless $\alpha$ is "controlled" by a small number of its terms; here $|\alpha|$ denotes the number of…

Let $\mathbb{F}_qG$ be a finite group algebra. We denote by $P(\mathbb{F}_qG)$ the probability that the product of two elements of $\mathbb{F}_qG$ be zero. In this paper, the general formula for computing the $P(\mathbb{F}_qG)$ are…

We give several examples of finite groups $G$ for which the rank of the tensor product $\mathbb{Z} \otimes_{\mathbb{Z}\mathrm{Aut}(G)}$ Wh$(G)$ is or is not zero. This is motivated by an earlier theorem of the first author, which implies as…

Let $G$ be a finite group, and $c$ an element of $\mathbb{Z}\cup \{\infty\}$. A subgroup $H$ of $G$ is said to be {\it $c$-nilpotent} if it is nilpotent, and has nilpotency class at most $c$. A subset $X$ of $G$ is said to be {\it…

Fix $k \geq 6$. We prove that any large enough finite group $G$ contains $k$ elements which span quadratically many triples of the form $(a,b,ab) \in S \times G$, given any dense set $S \subseteq G \times G$. The quadratic bound is…

Let $G$ be a finite group, define $I(G)=\{x\in G : x^{2}=1\}$, $C(G)=$ set of the cyclic subgroups of $G$, $i(G)=|I(G)|$ and $c(G)=|C(G)|$. In this article, we will classify finite groups with $i(G)=c(G)-r$ for $r=0,1,$ and $2$. We also…

Suppose that $X$ be a nonempty set. Denote by $\mathcal{T}(X)$ the full transformation semigroup on $X$. For $\varnothing \neq Z\subseteq Y\subseteq X$, let $\mathcal{T}(X,Y,Z)=\{\alpha \in \mathcal{T}(X): Y\alpha \subseteq Z \}$. Then…

Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the equality…

We say that $(x,y,z)\in Q^3$ is an associative triple in a quasigroup $Q(*)$ if $(x*y)*z=x*(y*z)$. Let $a(Q)$ denote the number of associative triples in $Q$. It is easy to show that $a(Q)\ge |Q|$, and we call the quasigroup maximally…

Let $G$ be a finite simple group of Lie type and let $P$ be a Sylow $2$-subgroup of $G$. In this paper, we prove that for any nontrivial element $x \in G$, there exists $g \in G$ such that $G = \langle P, x^g \rangle$. By combining this…

Let a real Lie group $G$ have a $C^\infty$ action on a real manifold $M$. Assume every nontrivial element of $G$ has nowhere dense fixpoint set in $M$. First, we show, in every frame bundle, except possibly the $0$th, that each stabilizer…

A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)$ implies that $C(x) = C(y)$. On the otherhand, two elements of a group are said to be $z$-equivalent or in the same $z$-class if their…

For a given free group $F$ of arbitrary rank (possibly infinite), and its subgroup $G$, we address the question whether a lower central subgroup of $G$ can contain a lower central subgroup of $F$. We show that the answer is no if $G$ does…

Let $G$ be a finite group and $S$ a subset of $G$. Then $S$ is {\em product-free} if $S \cap SS = \emptyset$, and $S$ {\em fills} $G$ if $G^{\ast} \subseteq S \cup SS$. A product-free set is locally maximal if it is not contained in a…