Splitting of Sharply 2-Transitive Groups of Characteristic 3

The character table of a sharply 5-transitive subgroup of ${\rm Alt}(12)$

In this paper we calculate the character table of a sharply $5$-transitive subgroup of ${\rm Alt}(12)$, and of a sharply $4$-transitive subgroup of ${\rm Alt}(11)$. Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups, and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations.

奇正特征下的不可分裂的锐2传递群

Non-split sharply 2-transitive groups of odd positive characteristic

本文构造了奇正特征下首个不可分裂的锐2传递群，并证明无2阶挠元的群可嵌入此类群，从而在足够大的特征下得到大量不同构的例子。

On the geometry of sharply 2-transitive groups

We show that the geometry associated to certain non-split sharply 2-transitive groups does not contain a proper projective plane. For a sharply 2-transitive group of finite Morley rank we improve known rank inequalities for this geometry and conclude that a sharply 2-transitive group of Morley rank 6 must be of the form $K\rtimes K^*$ for some algebraically closed field $K$.

Sharply transitive sets in $\mathrm{PGL}_2(K)$

Here is a simplified proof that every sharply transitive subset of $\mathrm{PGL}_2(K)$ is a coset of a subgroup.

A sharply 2-transitive group without a non-trivial abelian normal subgroup

We show that any group $G$ is contained in some sharply 2-transitive group $\mathcal{G}$ without a non-trivial abelian normal subgroup. This answers a long-standing open question. The involutions in the groups $\mathcal{G}$ that we construct have no fixed points.

Non-split linear sharply $2$-transitive groups

We give examples of countable linear groups in $SL_{n}(R)$ for $n \ge 3$, with no nontrivial normal abelian subgroups, that admit a faithful sharply 2-transitive action on a set. Without the linearity assumption, such groups were recently constructed by Rips, Segev, and Tent. Our examples are of permutational characteristic $2$, in the sense that involutions do not fix a point in the $2$-transitive action.

On correspondence between right near-domains and sharply 2-transitive groups

The right near-domain is defined to loosen near-domain axioms. Correspondence of a class of the right near-domains and a class of sharply 2--transitive groups is constructed.

有限生成的单的锐2-传递群

Finitely generated simple sharply 2-transitive groups

本文构造了首个有限生成的无限锐2-传递群，并进一步构造了一个具有Kazhdan性质(T)、单且恰有四个共轭类的此类群，同时证明该共轭类数量已是最小可能。

非分裂的 sharply 2-和3-传递群在SL_n(Z)中

Non-split sharply 2- and 3-transitive groups in SL_n(\mathbb Z)

本文研究了SL_3(Z)中非分裂的sharply 2-传递子群的存在性，并证明SL_4(Z)中存在非分裂的sharply 3-传递子群，但SL_3(Z)中不存在无限的sharply 3-传递子群。

Sharply 2-transitive linear groups

A group G is sharply 2-transitive if it admits a faithful permutation representation that is transitive and free on pairs of distinct points. Conjecturally, for all such groups there exists a near-field N (i.e. a skew field that is distributive only from the left) such that G is isomorphic to the semidirect product of the multiplicative and additive groups of N. This is well known in the finite case. We prove this conjecture when G < GL(n,F) is a linear group. Here we have to assume that both the characteristic of the field F and the permutational characteristic of the group G (see Definition …

有界指数的非分裂严格2-传递群

Non-split sharply 2-transitive groups of bounded exponent

本文构造了奇正特征下首例有界指数的非分裂严格2-传递群，并证明了存在可数无穷多个此类群，同时回答了关于2-传递置换群的两个公开问题。

On local sharply n-transitive groups

The paper is devoted to generalizations of actions of topological groups on manifolds. Instead of a topological group, we consider a local topological group generalizing the notion of a~germ or a~neighborhood in a topological group. The notion of an action of a local group on a topological space is introduced. The paper constructs the theory of local sharply $n$-transitive groups and local $n$-pseudofields. Local sharply $n$-transitive groups are reduced to simpler algebraic objects -- local $n$-pseudofields, similarly to the way Lie groups are reduced to Lie algebras, and sharply two-transiti…

On the non-existence of sharply transitive sets of permutations in certain finite permutation groups

In this short note we present a simple combinatorial trick which can be effectively applied to show the non--existence of sharply transitive sets of permutations in certain finite permutation groups.

The non-existence of sharply $2$-transitive sets of permutations in $\mathrm{Sp}(2d,2)$ of degree $2^{2d-1} \pm 2^{d-1}$

We use Müller and Nagy's method of contradicting subsets to give a new proof for the non-existence of sharply $2$-transitive subsets of the symplectic groups $\mathrm{Sp}(2d,2)$ in their doubly-transitive actions of degrees $2^{2d-1}\pm 2^{d-1}$. The original proof by Grundhöfer and Müller was rather complicated and used some results from modular representation theory, whereas our new proof requires only simple counting arguments.

Infinite sharply multiply transitive groups

The finite sharply $2$-transitive groups were classified by Zassenhaus in the 1930's. They essentially all look like the group of affine linear transformations $x\mapsto ax+b$ for some field (or at least near-field) $K$. However, the question remained open whether the same is true for infinite sharply $2$-transitive groups. There has been extensive work on the structures associated to such groups indicating that Zassenhaus' results might extend might extend to the infinite setting. For many specific classes of groups, like Lie groups, linear groups, or groups definable in o-minimal it was inde…

关于尖锐2传递群与近域的若干注记

Some remarks on sharply 2-transitive groups and near-domains

本文证明特征0的尖锐2传递群在点稳定子含有限指数阿贝尔子群时可分裂，并推广至近域，回答Kourovka笔记本问题12.48 b)。

关于非分裂特征2的严格k-传递作用及其推广的简单 acylindrical 构造方法

A simple acylindrical recipe for non-split characteristic $2$ sharply $k$-transitive actions and their generalizations

该研究给出群在δ-双曲空间上 acylindrical 作用的条件，可构造出k-严格传递作用，还得到有限生成的分裂严格2-、3-传递作用的简单例子。

Groups of finite Morley rank with a generically sharply multiply transitive action

We prove that if $G$ is a group of finite Morley rank which acts definably and generically sharply $n$-transitively on a connected abelian group $V$ of Morley rank $n$ with no involutions, then there is an algebraically closed field $F$ of characteristic $\ne 2$ such that $V$ has a structure of a vector space of dimension $n$ over $F$ and $G$ acts on $V$ as the group $\operatorname{GL}_n(F)$ in its natural action on $F^n$. This is the final pre-publication version of the paper: A. Berkman and A. Borovik, Groups of finite Morley rank with a generically sharply multiply transitive action, J. Alg…

当q≥13时，SL(2,q)不存在强传递子集

There are no sharply transitive subsets of $\mathrm{SL}(2,q)$ for $q\ge 13$

该研究将SL(2,q)强传递子群的存在性结论推广到强传递子集，证明当q≥13时SL(2,q)不存在此类子集，仅q∈{2,3,5,7,11}时存在。

Fissioned triangular schemes via sharply 3-transitive groups

n [D. de Caen, E.R. van Dam. Fissioned triangular schemes via the cross-ratio, {Europ. J. Combin.}, 22 (2001) 297-301], de Caen and van Dam constructed a fission scheme $\FT(q+1)$ of the triangular scheme on $\PG(1,q)$. This fission scheme comes from the naturally induced action of $\PGL(2,q)$ on the 2-element subsets of $\PG(1,q)$. The group $\PGL(2,q)$ is one of two infinite families of finite sharply 3-transitive groups. The other such family $\Mq(q)$ is a "twisted" version of $\PGL(2,q)$, where $q$ is an even power of an odd prime. The group $\PSL(2,q)$ is the intersection of $\PGL(2,q)$ a…

尖锐k-传递性的两种推广

Two generalisations of sharp k-transitivity

本文考虑群作用的尖锐\(k -\)传递性的两种推广。一是推广到尖锐\(\Theta -\)传递作用，刻画了相关条件及作用群性质，给出\(k = 2, 3\)时非分裂有限表示群例子并强化Tits结果；二是推广到关系结构的尖锐\(k -\)齐次作用，回答了Cameron的问题。

Affine Extensions of loops

We show a simple geometric procedure for an extension of a loop realized as the image $Σ^{\ast}$ of a sharply transitive section in a subgroup $G^{\ast}$ of the projective linear group $PGL(n-1, \mathbb K)$ to a loop realized as the image of a sharply transitive section in a group $Δ=T' \rtimes C$ of affinities of the $n$-dimensional space ${\cal A}_n=\mathbb K^n$ over a commutative field $\mathbb K$. We desire that $T'$ is a large subgroup of affine translations and that $α(C)=G^{\ast}$ holds for the canonical homomorphism $α:GL(n,\mathbb K) \to PGL(n, \mathbb K)$. We demonstrate that our con…

Locally compact convergence groups and n-transitive actions

All sigma-compact, locally compact groups acting sharply n-transitively and continuously on compact spaces M have been classified, except for n=2,3 when M is infinite and disconnected. We show that no such actions exist for n=2 and that these actions for n=3 coincide with the action of a hyperbolic group on a space equivariantly homeomorphic to its hyperbolic boundary. We further give a characterization of non-compact groups acting 3-properly and transitively on infinite compact sets as non-elementary boundary transitive hyperbolic groups. The main tool is a generalization to locally compact g…

A categorical approach to loops, neardomains and nearfields

In this paper we study loops, neardomains and nearfields from a categorical point of view. By choosing the right kind of morphisms, we can show that the category of neardomains is equivalent to the category of sharply 2-transitive groups. The other categories are also shown to be equivalent with categories whose objects are sets of permutations with suitable extra properties. Up to now the equivalence between neardomains and sharply 2-transitive groups was only known when both categories were equipped with the obvious isomorphisms as morphisms. We thank Hubert Kiechle for this observation.

再论群-子群子因子

Group-subgroup subfactors revisited

本文重新研究群-子群子因子的分类问题，证明对Frobenius群及一大类有限多重传递置换群（含锐k-传递群和M_{11}），其子因子可完全由主图唯一刻画。

Division algebras and transitivity of group actions on buildings

Let D be a division algebra with center F and degree d>2. Let K|F be any splitting field. We analyze the action of D^\times and SL_1(D) on the spherical and affine buildings that may be associated to GL_d(K) and SL_d(K), and in particular show it is never strongly transitive. In the affine case we find examples where the action is nonetheless Weyl transitive. This extends results of Abramenko and Brown concerning the d=2 case, where strong transitivity is in fact possible. Our approach produces some explicit constructions, and we find that for d>2 the failure of the action to be strongly trans…

泛型锐传递性的次数界定

Bounding the degree of generic sharp transitivity

本研究在L-群条件下证明了有限Morley秩泛型锐t-传递置换群的次数满足t≤r+2，推进了Borovik-Cherlin猜想，证明基于三个独立的群作用相关关键要素。

Trees of cylinders and canonical splittings

Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders T_c. This tree only depends on the deformation space of T; in particular, it is invariant under automorphisms of G if T is a JSJ splitting. We thus obtain Out(G)-invariant cyclic or abelian JSJ splittings. Furthermore, T_c has very strong compatibility properties (two trees are compatible if they have a…

Simply transitive NIL-affine actions of solvable Lie groups

Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can act simply transitive on which Lie groups $H$. So far the focus was mainly on the case where $G$ is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given ac…

相对自由分裂与自由因子复形I：双曲性

Relative free splitting and free factor complexes I: Hyperbolicity

研究自由群$F_n$及一般群$\Gamma$相对于自由因子系统选择的相对自由分裂复形和相对自由因子复形的大规模几何，证明其双曲性，证明还给出复形中粗传递拟测地线族信息。

Hyperfield extensions, characteristic one and the Connes-Consani plane connection

Inspired by a recent paper of Alain Connes and Catherina Consani which connects the geometric theory surrounding the elusive field with one element to sharply transitive group actions on finite and infinite projective spaces ("Singer actions"), we consider several fudamental problems and conjectures about Singer actions. Among other results, we show that virtually all infinite abelian groups and all (possibly infinitely generated) free groups act as Singer groups on certain projective planes, as a corollary of a general criterion. We investigate for which fields $\mathbb{F}$ the plane $\mathbf…