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Let $H$ and $B$ be subgroups of a finite group $G$ such that $G=N_{G}(H)B$. Then we say that $H$ is \emph{quasipermutable} (respectively \emph{$S$-quasipermutable}) in $G$ provided $H$ permutes with $B$ and with every subgroup (respectively…

Given a graph $G$, a subset $M$ of $V(G)$ is a module of $G$ if for each $v\in V(G)\setminus M$, $v$ is adjacent to all the elements of $M$ or to none of them. For instance, $V(G)$, $\emptyset$ and $\{v\}$ ($v\in V(G)$) are modules of $G$…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$, $G$ a finite group and $\sigma (G) =\{\sigma_{i} |\sigma_{i}\cap \pi (G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$ is said to be a…

In this paper, we prove a series of results on group embeddings in groups with a small number of generators. We show that each finitely generated group $G$ lying in a variety ${\mathcal M}$ can be embedded in a $4$-generated group $H \in…

For a sequence $S$ of terms from an abelian group $G$ of length $|S|$, let $\Sigma_n(S)$ denote the set of all elements that can be represented as the sum of terms in some $n$-term subsequence of $S$. When the subsum set is very small,…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this paper, we prove that if $\sigma_1(G)<2+\frac{11}{|G|}$\,, then $G$ is supersolvable. In particular, some new characterizations of the well-known groups…

Implicational bases are objects of interest in formal concept analysis and its applications. Unfortunately, even the smallest base, the Duquenne-Guigues base, has an exponential size in the worst case. In this paper, we use results on the…

We prove that if $(H,G)$ is a small, $nm$-stable compact $G$-group, then $H$ is nilpotent-by-finite, and if additionally $\NM(H) \leq \omega$, then $H$ is abelian-by-finite. Both results are significant steps towards the proof of the…

We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the problem to primitive actions of almost simple groups of Lie…

Let $g$ be an element of a finite group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots,g]$ over $x\in G$, where $g$ is repeated $n$ times. By Baer's theorem, if $E_n(g)=1$,…

Consider $\operatorname{Sym}(n)$, endowed with the normalized Hamming metric $d_n$. A finitely-generated group $\Gamma$ is \emph{P-stable} if every almost homomorphism $\rho_{n_k}\colon \Gamma\rightarrow\operatorname{Sym}(n_k)$ (i.e., for…

We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,Y^G) = G for a random subset Y of X, tends to 1 as n tends to infinity. So the property of the title…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this note, we prove that if $\sigma_1(G)<\frac{117}{20}$, then $G$ is solvable. Moreover, we have $\sigma_1(G)=\frac{117}{20}$ if and only if $G\cong A_5$.…

It has been proved in \cite{ge} for every $p$-group of order $p^n$, $|\mathcal{M}(G)|=p^{\f{1}{2}n(n-1)-t(G)}$, where $t(G)\geq 0$. In \cite{be, el, zh}, the structure of $G$ has been characterized for $t(G)=0,1,2,3$ by several authors.…

For the class of solvable groups of homeomorphisms of the line preserving orientation and containing a freely acting element, we establish the metabelianity of the quotient group $G/H_G$, where the elements of the normal subgroup $H_G$ are…

In the field of algebraic systems biology, the number of minimal polynomial models constructed using discretized data from an underlying system is related to the number of distinct reduced Gr\"obner bases for the ideal of the data points.…

We explore the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse and other) defined by the size of their elements. Applying the obtained results to the Stone-\v{C}ech compactification $\beta…

An algebraic structure is said to be congruence permutable if its arbitrary congruences $\alpha$ and $\beta$ satisfy the equation $\alpha \circ \beta =\beta \circ \alpha$, where $\circ$ denotes the usual composition of binary relations. For…

If a group $G$ is the union of proper subgroups $H_1, \dots, H_k$, we say that the collection $\{H_1, \dots H_k \}$ is a cover of $G$, and the size of a minimal cover (supposing one exists) is the covering number of $G$, denoted…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of $G$, we prove that if $\sigma_1(G)<\frac{117}{20}\,$, then $G$ is…