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Let $G$ be a finite almost simple group with socle $G_0$. A (nontrivial) factorization of $G$ is an expression of the form $G=HK$, where the factors $H$ and $K$ are core-free subgroups. There is an extensive literature on factorizations of…

We conjecture that if $G$ is a finite primitive group and if $g$ is an element of $G$, then either the element $g$ has a cycle of length equal to its order, or for some $r,m$ and $k$, the group $G\leq S_m\wr S_r$, preserving a product…

This is the fourth paper in a series. We prove a conjecture made independently by Boston et al and Shalev. The conjecture asserts that there is an absolute positive constant delta such that if G is a finite simple group acting transitively…

Let $G$ be a transitive permutation group on $\Omega$ with two points $\alpha, \beta\in\Omega$ such that $G_{\alpha}\cap G_{\beta}=1$. The Saxl graph $\Sigma(G)$ of the pair $(G,\Omega)$ is the graph with vertex set $\Omega$, while two…

Let $H$ be a subgroup of a group $G$. The permutizer $P_G(H)$ is the subgroup generated by all cyclic subgroups of $G$ which permute with $H$. A subgroup $H$ of a group $G$ is strongly permutable in $G$ if $P_U(H)=U$ for every subgroup $U$…

Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering of the finite group $G$. We prove that if $G$ is a group with $\lambda(G) \leqslant 6$, then $G$ is supersolvable. We also describe the structure of the groups…

We give an affirmative answer to the Grunwald problem for new families of non-solvable finite groups G, away from the set of primes dividing |G|. Furthermore, we show that such G verify the condition (BM), that is, the Brauer-Manin…

In this paper we describe some properties of groups $G$ that contain a solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2--3). We prove that if $G$ is a non-solvable group that contains a solvable subgroup of index…

We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble…

Guralnick, Kunyavskii, Plotkin and Shalev have shown that the solvable radical of a finite group $G$ can be characterized as the set of all $x\in G$ such that $<x,y>$ is solvable for all $y\in G$. We prove two generalizations of this…

The subdegrees of a transitive permutation group are the orbit lengths of a point stabilizer. For a finite primitive permutation group which is not cyclic of prime order, the largest subdegree shares a non-trivial common factor with each…

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X| bounded is…

Suppose that a group $G$ has socle $L$ a simple large-rank classical group. Suppose furthermore that $G$ acts transitively on the set of lines of a linear space $\mathcal{S}$. We prove that, provided $L$ has dimension at least 25, then $G$…

For a finite group $G$, let $\sigma(G)$ be the number of subgroups of $G$ and $\sigma_\iota(G)$ the number of isomorphism types of subgroups of $G$. Let $L=L_r(p^e)$ denote a simple group of Lie type, rank $r$, over a field of order $p^e$…

The solvability of monomial groups is a well-known result in character theory. Certain properties of Artin L-series suggest a generalization of these groups, namely to such groups where every irreducible character has some multiple which is…

For a simple linear algebraic group $G$ acting faithfully on a vector space $V$ and under mild assumptions, we show: if $V$ is large enough, then the Lie algebra of $G$ acts generically freely on $V$. That is, the stabilizer in the Lie…

Let p be a prime larger than 3 and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at most two distinct character degrees relatively prime to p in the principal p-block of G. This generalizes a…

A. Smoktunowicz and L. Vendramin conjectured that if $A$ is a finite skew brace with solvable additive group, then the multiplicative group of $A$ is solvable. In this short note we make a step towards positive solution of this conjecture…

Let $G$ be a finite primitive permutation group on a set $\Omega$ with nontrivial point stabilizer $G_{\alpha}$. We say that $G$ is extremely primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha\}$.…

The involution fixity ${\rm ifix}(G)$ of a permutation group $G$ of degree $n$ is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type.…