Related papers: Simple sharply 2-transitive groups
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.
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…
It is well-known that every sharply 2-transitive group of characteristic 3 splits. Here we construct the first examples of non-split sharply 2-transitive groups in odd positive characteristic $p$, for sufficiently large primes $p$.…
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 examples of infinite sharply 2-transitive groups which are finitely generated. Moreover, we construct such a group that has Kazhdan property (T), is simple, has exactly four conjugacy classes, and we show that this…
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…
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
We give a group theoretic proof of the splitting of sharply 2-transitive groups of characteristic 3.
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…
We construct here the first known examples of non-split sharply 2-transitive groups of bounded exponent in odd positive characteristic for every large enough prime $p \equiv 3 \pmod{4}$. In fact, we show that there are countably many…
A sharply 2-transitive permutation group of finite Morley rank and characteristic 2 splits; a split sharply 2-transitive permutation group of finite Morley rank and characteristic different from 2 is the group of affine transformations of…
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…
Here is a simplified proof that every sharply transitive subset of $\mathrm{PGL}_2(K)$ is a coset of a subgroup.
A finite transitive permutation group is said to be 3/2-transitive if all the nontrivial orbits of a point stabilizer have the same size greater than 1. Examples include the 2-transitive groups, Frobenius groups and several other less…
We give a uniform construction that, on input of a recursive presentation $P$ of a group, outputs a recursive presentation of a torsion-free group, isomorphic to $P$ whenever $P$ is itself torsion-free. We use this to re-obtain a known…
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.
Using a characterization of parabolics in reductive Lie groups due to Furstenberg, elementary properties of buildings, and some algebraic topology, we give a new proof of Tits' classification of 2-transitive Lie groups.
A sharply 2-transitive permutation group of characteristic 0 whose point stabiliser has an abelian subgroup of finite index splits. More generally, a near-domain of characteristic 0 with a multiplicative subgroup of finite index avoiding…
We construct connected $2$-arc-transitive covers of complete graphs with non-abelian characteristically simple transformation groups. This solves the existence problem for non-solvable $2$-arc-transitive covers of complete graphs.
We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. This paper classifies those lines that exhibit almost simple symmetries. We introduce a general recipe…