Yair Glasner, Dennis D. Gulko
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.
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We construct simple sharply 2-transitive groups. Our result answers an open question of Peter Neumann. In fact, we prove that every sharply 2-transitive group of characteristic 0 embeds into a simple sharply 2-transitive group.
We give an explicit construction of sharply $2$-transitive groups with fixed point free involutions and without nontrivial abelian normal subgroup.
We construct the first sharply $3$-transitive groups not arising from a near field, i.e. point stabilizers have no nontrivial abelian normal subgroup.
A number of conditions were found under which a sharply doubly transitive permutation group has an abelian normal divider.
We construct sharply 2-transitive groups of characteristic 0 without non-trivial abelian normal subgroup. These groups act sharply 2-trnaisitvely by conjugation on their involutions. This answers a longstanding open question.
In this short note we show how to modify the construction of non-split sharply $2$-transitive groups of characteristic~$0$ given by Rips and Tent [arXiv:1604.00573] to allow for arbitrary fields of characteristic 0
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.
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。
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.
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。
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。
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。
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.
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…
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)。
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。
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…
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 $K$.
Here is a simplified proof that every sharply transitive subset of $\mathrm{PGL}_2(K)$ is a coset of a subgroup.
We give a group theoretic proof of the splitting of sharply 2-transitive groups of characteristic 3.
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.
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…
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.
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。
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 …
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…
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…
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。
We determine character tables for twisted fractional linear groups that form the `other' family in Zassenhaus' classification of finite sharply 3-transitive groups.
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…
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.
A regular covering projection $\p\colon \tX \to X$ of connected graphs is $G$-admissible if $G$ lifts along $\p$. Denote by $\tG$ the lifted group, and let $\CT(\p)$ be the group of covering transformations. The projection is called $G$-split whenever the extension $\CT(\p) \to \tG \to G$ splits. In this paper, split 2-covers are considered. Supposing that $G$ is transitive on $X$, a $G$-split cover is said to be $G$-split-transitive if all complements $\bG \cong G$ of $\CT(\p)$ within $\tG$ are transitive on $\tX$; it is said to be $G$-split-sectional whenever for each complement $\bG$ there …
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。
This paper contains the more significant part of the article with the same title that will appear in the Volume 12 of Journal of Group Theory (2009). In this paper we determine all algebraic transformation groups $G$, defined over an algebraically closed field $\sf k$, which operate transitively, but not primitively, on a variety $Ω$, provided the following conditions are fulfilled. We ask that the (non-effective) action of $G$ on the variety of blocks is sharply 2-transitive, as well as the action on a block $Δ$ of the normalizer $G_Δ$. Also we require sharp transitivity on pairs $(X,Y)$ of i…
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…
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。
Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank
该研究定义伪双曲反射空间概念，用于分析有限Morley秩的Frobenius群，得出相关群的分裂性结论与秩不等式推论。
Generalizing classical extension theory, we solve a Schreier-type extension problem for polygroups by groups. As a consequence, we obtain a method for computing a presentation for a group from its action on a set. The usefulness of this method is illustrated by deriving explicit presentations for the groups $GL_2$ over valuation rings and over valued fields, for the groups $SL_3$ over arbitrary fields, as well as for the five Mathieu groups. Moreover, we sketch some aspects of a new deformation technique for groups, their actions, and presentations, and apply it to compute presentations for th…
A hovel is a generalization of the Bruhat-Tits building that is associated to an almost split Kac-Moody group G over a non-Archimedean local field. In particular, G acts strongly transitively on its corresponding hovel $Δ$ as well as on the building at infinity of $Δ$, which is the twin building associated to G. In this paper we study strongly transitive actions of groups that act on affine ordered hovels $Δ$ and give necessary and sufficient conditions such that the strong transitivity of the action on $Δ$ is equivalent to the strong transitivity of the action of the group on its building at …
Elusive groups from non-split extensions
本文通过非分裂扩展构造了新的难以捉摸群，包括新的度数和奇数度数的例子，并提出进一步研究问题。
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…
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…
Let $Ω=\{1,2,...,n\}$ where $n \ge 2$. The {\em shape} of an ordered set partition $P=(P_1,..., P_k)$ of $Ω$ is the integer partition $λ=(λ_1,...,λ_k)$ defined by $λ_i = |P_i|$. Let G be a group of permutations acting on $Ω$. For a fixed partition $λ$ of n, we say that G is {\em $λ$-transitive} if G has only one orbit when acting on partitions P of shape $\la$. A corresponding definition can also be given when G is just a set. For example, if $λ=(n-t,1,...,1)$, then a $λ$-transitive group is the same as a t-transitive permutation group and if $λ=(n-t,t)$, then we recover the t-homogeneous perm…
Kernels in measurable cohomology for transitive actions
本文推广了 Monod 关于 Furstenberg 边界上可测上同调的结果，证明对满足几乎处处点对稳定化子紧的子群 $L<P$，$G$ 在 $G/L$ 上的可测上同调满射到 $G$ 的可测上同调，核为 $H^{p-1}_m(L)$，并给出具体例子与余循环计算。
An infinite family of counterexamples to the Polycirculant Conjecture
反驳多循环猜想，证明存在无半正则自同构的顶点传递图。采用Chen等人方法构造度为16464的难以捉摸群\(7^6.\mathrm{PSU}_3(3)\)，其是2 - 闭的，此例扩展出多循环猜想无穷反例及相关顶点传递图。
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…
Sharp Uncertainty Principle for Transitive $G$-Sets over Arbitrary Fields and Finite Groups
该研究将有限交换群的严格不确定性原理扩展到任意域和有限群的传递$G$-集框架，构造块和子集证明新不等式，给出等号条件，解决开放问题并恢复Tao的强不确定性原理。
Non-Wiener groups with a Gelfand pair
本文证明，若非顺从局部紧群G具有紧子群K使得(G,K)为Gelfand对，且G存在拓扑不可约且不可酉化的边界表示，则G不是Wiener群，即其傅里叶变换不满足Wiener Tauberian定理的类比。作为应用，证明了具有无穷多端的连通局部有限图的非紧边界传递自同构群，或非阿基米德局部域上的非交换分裂约化代数群，都不是Wiener群。
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