Nick Gill
A long-standing conjecture is that any transitive finite projective plane is Desarguesian. We make a contribution towards a proof of this conjecture by showing that a group acting transitively on the the points of a non-Desarguesianprojective plane must not contain any components.
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Suppose that a group $G$ acts transitively on the points of $\mathcal{P}$, a finite non-Desarguesian projective plane. We prove that if $G$ is insoluble then $G/O(G)$ is isomorphic to $SL_2(5)$ or $SL_2(5).2$.
Suppose that a group $G$ acts transitively on the points of a non-Desarguesian plane, $\mathcal{P}$. We prove first that the Sylow 2-subgroups of $G$ are cyclic or generalized quaternion. We also prove that $\mathcal{P}$ must admit an odd order automorphism group which acts transitively on the set of points of $\mathcal{P}$.
In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and…
This paper is a further contribution to the classification of line-transitive finite linear spaces. We prove that if S is a non-trivial finite linear space with the Fang-Li parameter gcd(k,r) is 9 or 10, the automorphism group G of S is line-transitive and point-imprimitive, then S is the Desarguesian projective plane PG(2,9).
Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $π$ acts $2$-transitively on the points of $π$, then $π$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $\mathrm{PΓL}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible spherical building, also known as a \emph{generalized polygon}, the theorem of Fong and Seitz (1973) gave a classification of the \emph{Moufang} examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-tra…
Block-transitive t-(k^2,k,λ) designs associated to two dimensional projective special linear groups
本文研究区传递自同构群作用下的 t-(k^2,k,λ) 设计，证明当正规子群为 PSL(2,q)（q≥4）时，仅当 q=8 时存在非平凡设计，且必为参数 λ∈{2,6,9,12,18,36} 的 2-(36,6,λ) 设计。
In this article, we study symmetric $(v, k, λ)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is a projective special unitary group of dimension at most five. We, in particular, determine all such possible parameters $(v, k, λ)$ and show that there exist eight non-isomorphic of such designs for which $λ\in\{3,6,12, 16, 18\}$ and $G$ is $PSU_{3}(3)$, $PSU_{3}(3):2$, $PSU_{4}(2)$ or $PSU_{4}(2):2$.
Given a finite group $G$. The generating pair $(H,a)$ of $G$, that is, $H<G$ and $a\in G$ such that $\langle a,H\rangle=G$. In this paper, we introduce the definition of FF-subgroup to characterize the generating pairs of the symmetric groups, alternating groups and projective groups $PSL(2,q)$. This gives a partial answer to an open problem of J. André and J. Araújo and P. J. Cameron.
From the point of view of discrete geometry, the class of locally finite transitive graphs is a wide and important one. The subclass of Cayley graphs is of particular interest, as testifies the development of geometric group theory. Recall that Cayley graphs can be defined as non-empty locally finite connected graphs endowed with a transitive group action such that any non-identity element acts without fixed point. We define a class of transitive graphs which are transitive in an "absolutely non-Cayley way": we consider graphs endowed with a transitive group action such that any element of the…
We show that the only transitive and generically $4$-transitive action of a group of finite Morley rank on a set of Morley rank $2$ is the natural action of $\operatorname{PGL}_3$ on the projective plane.
Soft planes and groups
本文利用满足简单条件的群构造有限射影平面，所得平面包含许多已知平面并可能包含新平面，且精确刻画为具有固定旗并在特定旗集上传递的共线群的射影平面。
The Borovik-Cherlin conjecture holds in ACF
该研究证明了在代数闭域（ACF）中，连通群对不可约簇的特定置换作用同构于射影线性群的自然作用，确立了对应置换群的Borovik-Cherlin猜想。
Vertex-transitive graphs with uniformly bisecting quasi-geodesics
本文研究满足一致平分拟测地线性质的顶点可递图，证明其拟等距于欧氏或双曲平面，对应群为虚曲面群，正面解决了Kourovka笔记本的一个问题。
Regular algebraic surfaces isogenous to a higher product of curves can be obtained from finite groups with ramification structures. We find unmixed ramification structures for finite groups constructed as p-quotients of particular infinite groups with special presentations related to finite projective planes.
We classify equivariant topological complex vector bundles over real projective plane under a compact Lie group (not necessarily effective) action. It is shown that nonequivariant Chern classes and isotropy representations at (at most) three points are sufficient to classify equivariant vector bundles over real projective plane except one case. To do it, we relate the problem to classification on two-sphere through the covering map because equivariant vector bundles over two-sphere have been already classified.
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