Related papers: On the classification of extremely primitive affin…

This paper presents a classification of exact factorizations of almost simple groups, which has been a long-standing open problem initiated around 1980 by the work of Wiegold-Williamson, and significantly progressed by Liebeck, Praeger and…

This paper is a continuation of Almost Commutative Terwilliger Algebras of Group Association Schemes I: Classification [1]. In that paper, we found all groups G for which the Terwilliger algebra of the group association scheme, denoted T…

We classify the finite primitive permutation groups which have a cyclic subgroup with two orbits. This extends classical topics in permutation group theory, and has arithmetic consequences. By a theorem of C. L. Siegel, affine algebraic…

A nonnegative matrix $A$ is called primitive if $A^k$ is positive for some integer $k>0$. A generalization of this concept to finite sets of matrices is as follows: a set of matrices $\mathcal M = \{A_1, A_2, \ldots, A_m \}$ is primitive if…

The problem of describing the invariance groups of unordered relations, called briefly \emph{relation groups}, goes back to classical work by H. Wielandt. In general, the problem turned out to be hard, and so far it has been settled only…

We prove that an infinite family of semiprimitive groups are graph-restrictive. This adds to the evidence for the validity of the PSV Conjecture and increases the minimal imprimitive degree for which this conjecture is open to 12. Our…

General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point…

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer is trivial; the base size of $G$ is the minimal cardinality of a base. In this paper we initiate the study of bases for…

Finite W-algebras are certain associative algebras arising in Lie theory. Each W-algebra is constructed from a pair of a semisimple Lie algebra g (our base field is algebraically closed and of characteristic 0) and its nilpotent element e.…

A lattice-ordered group (an $\ell$-group) $G(\oplus, \vee, \wedge)$ can be naturally viewed as a semiring $G(\vee,\oplus)$. We give a full classification of (abelian) $\ell$-groups which are finitely generated as semirings, by first showing…

Many open conjectures in the representation theory of finite groups can be studied by reducing them to related questions about quasi-simple groups. In such studies, $p$-radical subgroups typically play a critical role. To classify the…

A complete mapping of a group $G$ is a bijection $\phi\colon G\to G$ such that $x\mapsto x\phi(x)$ is also bijective. Hall and Paige conjectured in 1955 that a finite group $G$ has a complete mapping whenever $\prod_{x\in G} x$ is the…

Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of…

Motivated by the maximal subgroup problem of the finite classical groups we begin the classification of imprimitive irreducible modules of finite quasisimple groups. We obtain our strongest results for modules over fields of characteristic…

We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. For every finite group $G$, a word $w$ in the free group on $k$ generators induces…

Let $G$ be a finite almost simple group and let $H$ be a Sylow $p$-subgroup of $G$. As a special case of a theorem of Zenkov, there exist $x,y \in G$ such that $H \cap H^x \cap H^y = 1$. In fact, if $G$ is simple, then a theorem of Mazurov…

A proper subgroup $H$ in a finite group $G$ is said to be large if $|H|^3\geq |G|$. In this paper, we determined all large maximal subgroups of almost simple classical groups. Combined with the work of Alavi and Burness (J. Algebra 421…

This paper is about the structure of infinite primitive permutation groups and totally disconnected locally compact groups ("tdlc groups'"). The permutation groups we investigate are subdegree-finite (i.e. all orbits of point stabilisers…

We say that a group $G$ is almost Engel if for every $g\in G$ there is a finite set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$, that is, for every…

Let $A$ be a $d\times n$ integer matrix whose column vectors generate the lattice $\Z^d$, and let $D(R_A)$ be the ring of differential operators on the affine toric variety defined by $A$. We show that the classification of…