相关论文: Conjugate generation of sporadic almost simple gro…

We show that if $G$ is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous ergodic invariant random…

For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group,…

We construct explicit geometric models for and compute the fundamental groups of all plane sextics with simple singularities only and with at least one type $\bold E_8$ singular point. In particular, we discover four new sextics with…

The spectrum of a group is the set of its element orders. A finite group $G$ is said to be recognizable by spectrum if every finite group that has the same spectrum as $G$ is isomorphic to $G$. We prove that the simple alternating groups…

We prove that the product of a subset and a normal subset inside any finite simple non-abelian group $G$ grows rapidly. More precisely, if $A$ and $B$ are two subsets with $B$ normal and neither of them is too large inside $G$, then $|AB|…

Let $\Gamma$ be a countably infinite discrete group. A $\Gamma$-flow $X$ (i.e., a nonempty compact Hausdorff space equipped with a continuous action of $\Gamma$) is called $S$-minimal for a subset $S \subseteq \Gamma$ if the partial orbit…

Given an action of a monoid $T$ on a ring $A$ by ring endomorphisms, and an Ore subset $S$ of $T$, a general construction of a fractional skew monoid ring $S^{\rm op} * A * T$ is given, extending the usual constructions of skew group rings…

It is a simple fact that a group has a trivial automorphism group if and only if it is of order $1$ or $2$. We prove that the same holds for certain families of skew braces, and given any odd prime $p$, we construct a skew brace of order…

Let $G = K \rtimes \langle t \rangle $ be a finitely generated group where $K$ is abelian and $\langle t\rangle$ is the infinite cyclic group. Let $ R $ be a finite symmetric subset of $K$ such that $S = \{ (r,1),(0,t^{\pm 1}) \mid r \in R…

A generating function of the number of homomorphisms from the fundamental group of a compact oriented or non-orientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate…

A numerical set $S$ is a cofinite subset of $\mathbb{N}$ which contains $0$. We use the natural bijection between numerical sets and Young diagrams to define a numerical set $\widetilde{S}$, such that their Young diagrams are complements.…

A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si > for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response…

A trivial automorphism of the Boolean algebra $\mathcal P(\mathbb N) / \mathrm{Fin}$ is an automorphism induced by the action of some function $\mathbb N \rightarrow \mathbb N$. In models of forcing axioms all automorphisms are trivial, and…

We classify the pairs $(X,\pi)$, where $\pi\colon X\to S$ is a $\mathbb{P}^1$-bundle over a non-rational geometrically ruled surface $S$ and $\mathrm{Aut}^\circ(X)$ is relatively maximal, i.e., maximal with respect to the inclusion in the…

We describe all groups that can be generated by two twists along spherical sequences in an enhanced triangulated category. It will be shown that with one exception such a group is isomorphic to an abelian group generated by not more than…

We describe the collection of finite simple groups, with a view on physical applications. We recall first the prime cyclic groups $Z_p$, and the alternating groups $Alt_{n>4}$. After a quick revision of finite fields $\mathbb{F}_q$, $q =…

We characterise when the pure monomorphisms in a presheaf category $\mathbf{Set}^\mathcal{C}$ are cofibrantly generated in terms of the category $\mathcal{C}$. In particular, when $\mathcal{C}$ is a monoid $S$ this characterises cofibrant…

The classical notion of a k-automatic subset of the natural numbers is here extended to that of an F-automatic subset of an arbitrary finitely generated abelian group $\Gamma$ equipped with an arbitrary endomorphism F. This is applied to…

Let $G$ be an almost simple sporadic group and let $H$ be a soluble subgroup of $G$. In this paper we prove that there exists $x,y \in G$ such that $H \cap H^x \cap H^y=1$, which is equivalent to the bound $b(G,H) \leqslant 3$ with respect…

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements…