Related papers: Sharply 2-transitive groups of finite Morley rank
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…
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…
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.
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 give an explicit construction of sharply $2$-transitive groups with fixed point free involutions and without nontrivial abelian normal subgroup.
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…
Several structural results about permutation groups of finite rank definable in differentially closed fields of characteristic zero (and other similar theories) are obtained. In particular, it is shown that every finite rank definably…
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 a group theoretic proof of the splitting of sharply 2-transitive groups of characteristic 3.
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…
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…
We show that a generically sharply $t$-transitive permutation group of finite Morley rank on a set of rank $r$ satisfies $t\le r+2$ provided the pointwise stabilizer of a generic $(t-1)$-tuple is an $L$-group, which holds, for example, when…
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 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…
Let $G$ be an infinite simple group of finite Morley rank and of Pr\"{u}fer $2$-rank $1$ which admits a supertight automorphism $\alpha$ such that the fixed-point subgroup $C_G(\alpha^n)$ is pseudofinite for all integers $n > 0$. We prove…
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 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 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 a review of one of the lines in development of the theory of groups of finite Morley rank. These groups naturally appear in model theory as model-theoretic analogues of Galois groups, therefore their actions and their role as…
A number of conditions were found under which a sharply doubly transitive permutation group has an abelian normal divider.