Related papers: Binary permutation groups: alternating and classic…

A semigroup of binary relations (under composition) on a set $X$ is \emph{complemented} if it is closed under the taking of complements within $X\times X$. We resolve a 1991 problem of Boris Schein by showing that the class of finite unary…

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative…

Given a conjugacy class $\mathcal{C}$ in a group $G$ we define a new graph, $\Gamma(\mathcal{C})$, whose vertices are elements of $\mathcal{C}$; two vertices $g,h\in \mathcal{C}$ are connected in $\Gamma(\mathcal{C})$ if $[g,h]=1$ and…

A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either semiregular or transitive.The class of semiprimitive groups properly contains primitive groups, quasiprimitive groups and innately…

We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the…

This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We…

We show that a non-algebraic simple group of finite Morley rank with a definable representation over a field has no involutions, and otherwise resembles a bad group. In particular, the modern form of the Cherlin-Zilber alebaricity…

We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the…

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 first main result of this paper is that a finite transitive nonabelian characteristically simple subgroup of a wreath product in product action must lie in the base group of the wreath product. This allows us to characterize nonabelian…

In this paper we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. We give bounds on the order, base size, minimal degree, fixity, and chief length of an arbitrary…

In this paper we introduce and study the relative cyclic subgroup commutativity degrees of a finite group. We show that there is a finite group with $n$ such degrees for all $n \in \mathbb{N}^*\setminus \lbrace 2\rbrace$ and we indicate…

Groups, in which every subgroup containing some fixed primary cyclic subgroup has a complement, are investigated.

We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let $n\geq 5$ be an integer, $G$ a finite group, and let $\AAA$ and $\SSS^\pm$ denote the double…

For an arbitrary finite permutation group $G$, subgroup of the symmetric group $S_\ell$, we determine the permutations involving only members of $G$ as $\ell$-patterns, i.e., avoiding all patterns in the set $S_\ell \setminus G$. The set of…

We investigate the transfer of the Cohen-Macaulay property from a commutative ring to a subring of invariants under the action of a finite group. Our point of view is ring theoretic and not a priori tailored to a particular type of group…

A group is metabelian if its commutator subgroup is abelian. For finitely generated metabelian groups, classical commutative algebra, algebraic geometry and geometric group theory, especially the latter two subjects, can be brought to bear…

Working in a theory with an integer-valued dimension on interpretable sets, we classify pseudofinite definably primitive permutation groups acting on one-dimensional sets which satisfy a version of chain condition on centralizers and on…

Let $G$ be a nontrivial transitive permutation group on a finite set $\Omega$. By a classical theorem of Jordan, $G$ contains a derangement, which is an element with no fixed points on $\Omega$. Given a prime divisor $r$ of $|\Omega|$, we…

We prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group of a full shift. Hence, they are…