Centralizer algebras for an automorphism group of a finite incidence structure

A well-known result of Dembowski and Wagner (4) characterizes the designs of points and hyperplanes of finite projective spaces among all symmetric designs. By passing to a dual situation and approaching this idea from a different direction, we shall obtain common characterizations of finite projective and affine spaces. Our principal result is the following.Theorem 1.A finite incidence structure is isomorphic to the design of points and hyperplanes of a finite projective or affine space of dimension greater than or equal to4if and only if there are positive integers v, k, and y, with μ> 1and(μ– l)(v — k) ≠ (k—μ)2such that the following assumptions hold.(I)Every block is on k points, and every two intersecting blocks are on μ common points.(II)Given a point and two distinct blocks, there is a block containing both the point and the intersection of the blocks.(III)Given two distinct points p and q, there is a block on p but not on q.(IV)There are v points, and v– 2 ≧k>μ.

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Finite Incidence Structures with Orthogonality

Canadian Journal of Mathematics ◽

10.4153/cjm-1967-098-9 ◽

1967 ◽

Vol 19 ◽

pp. 1078-1083 ◽

Cited By ~ 3

Author(s):

F. A. Sherk

Keyword(s):

Incidence Structure ◽

Affine Planes ◽

Axiom A ◽

Incidence Structures ◽

Finite Incidence

An incidence structure consists of two sets of elements, called points and blocks, together with a relation, called incidence, between elements of the two sets. Well-known examples are inversive planes, in which the blocks are circles, and projective and affine planes, in which the blocks are lines. Thus in various examples of incidence structures, the blocks may have various interpretations. Very shortly, however, we shall impose a condition (Axiom A) which ensures that the blocks behave like lines. In anticipation of this, we shall refer to the set of blocks as the set of lines. Also, we shall employ the usual terminology of incidence, such as “lies on,” “passes through,” “meet,” “join.” etc.

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Groups $S_n \times S_m$ in construction of flag-transitive block designs

Glasnik Matematicki ◽

10.3336/gm.56.2.02 ◽

2021 ◽

Vol 56 (2) ◽

pp. 225-240

Author(s):

Snježana Braić ◽

◽

Joško Mandić ◽

Aljoša Šubašić ◽

Tanja Vojković ◽

...

Keyword(s):

Automorphism Group ◽

Block Design ◽

Incidence Structure ◽

Block Designs ◽

Existence Problem ◽

Transitive Automorphism ◽

Transitive Automorphism Group ◽

Point Set ◽

The Given

In this paper, we observe the possibility that the group $S_{n}\times S_{m}$ acts as a flag-transitive automorphism group of a block design with point set $\{1,\ldots ,n\}\times \{1,\ldots ,m\},4\leq n\leq m\leq 70$. We prove the equivalence of that problem to the existence of an appropriately defined smaller flag-transitive incidence structure. By developing and applying several algorithms for the construction of the latter structure, we manage to solve the existence problem for the desired designs with $nm$ points in the given range. In the vast majority of the cases with confirmed existence, we obtain all possible structures up to isomorphism.

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ON THE AUTOMORPHISM GROUP OF A FINITE P-GROUP WITH CYCLIC FRATTINI SUBGROUP

Mathematical Proceedings of the Royal Irish Academy ◽

10.3318/pria.2008.108.2.165 ◽

2008 ◽

Vol 108 (2) ◽

pp. 165-175 ◽

Cited By ~ 5

Author(s):

S. Fouladi ◽

A. R. Jamali ◽

R. Orfi

Keyword(s):

Automorphism Group ◽

Frattini Subgroup

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On the automorphism group of a symplectic half-flat 6-manifold

Forum Mathematicum ◽

10.1515/forum-2018-0137 ◽

2019 ◽

Vol 31 (1) ◽

pp. 265-273

Author(s):

Fabio Podestà ◽

Alberto Raffero

Keyword(s):

Automorphism Group ◽

Lie Algebra ◽

Group Action ◽

Cohomogeneity One ◽

Cohomogeneity One Action

Abstract We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat {\mathrm{SU}(3)} -structure has Abelian Lie algebra with dimension bounded by {\min\{5,b_{1}(M)\}} . Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on {T\mathbb{S}^{3}} which are invariant under a cohomogeneity one action of {\mathrm{SO}(4)} .

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ON EDGE-PRIMITIVE GRAPHS WITH SOLUBLE EDGE-STABILIZERS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788721000112 ◽

2021 ◽

pp. 1-28

Author(s):

HUA HAN ◽

HONG CI LIAO ◽

ZAI PING LU

Keyword(s):

Automorphism Group

Abstract A graph is edge-primitive if its automorphism group acts primitively on the edge set, and $2$ -arc-transitive if its automorphism group acts transitively on the set of $2$ -arcs. In this paper, we present a classification for those edge-primitive graphs that are $2$ -arc-transitive and have soluble edge-stabilizers.

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Affine cones over cubic surfaces are flexible in codimension one

Forum Mathematicum ◽

10.1515/forum-2020-0191 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Alexander Perepechko

Keyword(s):

Automorphism Group ◽

Open Subset ◽

Pezzo Surface ◽

Ample Divisor ◽

Transitive Action ◽

Del Pezzo Surface ◽

Codimension One ◽

Cubic Surfaces ◽

Special Automorphism ◽

Del Pezzo

AbstractLet Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

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