Über symmetrische Graphen vom grad fünf

Let G be a connected 1-transitive graph of valency five. It is shown that the order of a vertex stabilizer divides 5 · 3² · 2¹⁷. A theorem of A. Gardiner bounding the order of a vertex stabilizer of a 2-transitive graph of valency 1 + p,p prime, is reproved.     

Classification of 2-transitive symmetric designs

All symmetric designs are determined for which the automorphism group is 2-transitive on the set of points. To Prof. Noboru Ito, to commemorate his 60th birthday This research was supported in part by NSF Grant MCS 7903130-82.     

Benz planes of Dembowski type and groups of tangential projectivities

Using the tangential relation we introduce in Benz planes M of Dembowski type, which generalize the Benz planes over algebras of characteristic 2, the group ?? of tangential perspectivities. We prove that these groups have the same behaviour as the classical groups of projectivities if any tangential perspectivity is induced by an automorphism of M. As permutation groups of a circle onto itself the groups ?? essentially differs from the classical groups of projectivities. If M is a Laguerre plane of Dembowski type, then ?? is always sharply 3-transitive. For Minkowski planes of Dembowski type ?? is at least 2-transitive. If M is a finite Benz plane of order 2 ^s , then ?? is isomorphic to the group PGL ₂(2 ^s ) in its sharply 3-transitive representation.     

Sharply 1-transitive subsets of certain permutation groups

In this paper we give a method for constructing sharply 1-transitive permutation sets inside a finite permutation group with certain properties and we apply this method to obtain a family of sharply 1-transitive permutation subsets of the sharply 3-transitive permutation group M(p ^2f ) on PG(1, p ^2f ) for p ^f 1 (mod 4).Work supported by G.N.S.A.G.A. and M.P.I.     

A characterization of classical Minkowski planes over a perfect field of characteristic two

We give a new set of axioms defining the concept of (B^*)-plane (i.e. Minkowski plane without the tangency property) and we show that every (B^*)-plane in which a condition similar to the “Fano condition” of Heise and Karzel (see [5, § 3]) holds, is a Minkowski plane over a perfect field of characteristic two. In particular, every finite (B^*)-plane of even order is a Minkowski plane over a field. Consequences for strictly 3-transitive groups are derived from the preceding results; in particular, every strictly 3-transitive set of permutations of odd degree containing the identity is a protective group PGL₂(GF(2 ⁿ )) over a finite field GF(2 ⁿ , for some positive integer n.     

Varietà di Segre e ovoidi dello spazio polare Q+(7, q)

Summary In this paper, for q even, we construct an ovoid O ₃ and a spread S of the finite classical polar space Q⁺(7, q) determinated by a hyperbolic quadric Q⁺ of PG(7, q) such that there is a subgroup of PGO ₊ ⁸ (q) isomorphic to PGL₂(q ³), which maps O ₃ in itself and S in S and is 3-transitive on O ₃ and on S; for q>2, S is not a Desarguesian spread of Q⁺(7, q) and O ₃ is a Desarguesian ovoid.

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Varietà di Segre e ovoidi dello spazio polare Q⁺(7, q)

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Al Prof. Adriano Barlotti in occasione del suo 60^o compleanno     

Over the non-existence of sharply 3-transitive permutation sets containing sharply 2-transitive permutation subsets

The following results are proved: Let E be a finite set, ¦E¦>4, and let G be a sharply 3-transitive permutation set on E. Then G contains no subset which is a sharply 2-transitive permutation set on E (Theorem 1). In the case when G is a sharply 3-transitive permutation group which is also planar, the finiteness condition on E can be dropped (Theorem 2).Dedicated to G. Zappa on his 70th birthdayResearch done within the activity of GNSAGA of CNR, supported by the 40% grants of MPI.     

Über die Normalteiler scharf zweifach und scharf dreifach transitiver Permutationsgruppen

A sharply 2-transitive (3-transitive) groupT can be described by means of a neardomainF (a KT-field(F,)). We show, thatT has a least nontrivial normal subgroupA (S(F,)), ifF is a nearfield or if CharF 2. In this case the nontrivial normal subgroups ofT correspond bijectively with all normal subgroupsF ^* (the multiplicative group ofF) containing a setD (D(Q)). IfF is a nearfield or ifF has a suitable central element, then the group S(F,) is simple.

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Herrn Prof. Dr. Dr. h. c. Helmut Karzel zum 70. Geburtstag gewidmet     

Extension of Multiply Transitive Permutation Sets

Let G be a k-transitive permutation set on E and let E* = E∪{∞},∞ ? E; if G* is a (k: + 1)-transitive permutation set on E*, G* is said to be an extension of G whenever G _∝ ^* =G. In this work we deal with the problem of extending (sharply) k- transitive permutation sets into (sharply) (k + 1)-transitive permutation sets. In particular we give sufficient conditions for the extension of such sets; these conditions can be reduced to a unique one (which is a necessary condition too) whenever the considered set is a group. Furthermore we establish necessary and sufficient conditions for a sharply k- transitive permutation set (k ≥ 3) to be a group. Math. Subj. Class.: 20B20 Multiply finite transitive permutation groups 20B22 Multiply infinite transitive permutation groups     

Semisymmetric graphs from polytopes

Every finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope.     

A Classification of Countable Lower 1-transitive Linear Orders

This paper contains a classification of countable lower 1-transitive linear orders. This is the first step in the classification of countable 1-transitive trees given in Chicot and Truss (2009): the notion of lower 1-transitivity generalises that of 1-transitivity for linear orders, and it is essential for the structure theory of 1-transitive trees. The classification is given in terms of coding trees, which describe how a linear order is fabricated from simpler pieces using concatenations, lexicographic products and other kinds of construction. We define coding trees and show that a coding tree can be constructed from a lower 1-transitive linear order \((X, \leqslant )\) by examining all the invariant partitions on X. Then we show that a lower 1-transitive linear order can be recovered from a coding tree up to isomorphism.     

Tetravalent s-Transitive Graphs of Order 4p 2

Let s be a positive integer. A graph is s -transitive if its automorphism group is transitive on s-arcs but not on (s?+?1)-arcs. Let p be a prime. Zhou (Discrete Math 309:6081?C6086, 2009) classified tetravalent s-transitive graphs of order 4p. In this article a complete classification of tetravalent s-transitive graphs of order 4p ² is given.     

On the sharply 1-transitive subsets of PGL(2,pm)

If p is an odd prime and R is a sharply 1-transitive subset of PGL(2,p^m) which contains the identity but is not a group, then the subgroup generated by R is either PSL(2,p^m) or PGL(2,p^m).work done within the activity of G.N.S.A.G.A. and supported by the Italian Ministry of Public EducationDedicated to Professor Helmut Karzel on his 60^th birthday     

6-Transitive graphs

A connected graph is n-transitive if, whenever two n-tuples are isometric, there is an automorphism mapping the first to the second. It is shown that a 6-transitive graph is complete multipartite, or complete bipartite with a matching deleted, or a cycle, or one of three special graphs on 9, 12 and 20 vertices. These graphs are n-transitive for all n; but there are graphs (the smallest on 56 vertices) which are 5- but not 6-transitive.     

The nonexistence of 8-transitive graphs

We prove that the inequalitys≦7 holds for finites-transitive graphs assuming that the list of known 2-transitive permutation groups is complete.     

MAXIMAL SUBGROUPS OF SL(<Emphasis Type="Italic">n</Emphasis>, ℤ)

We establish the existence of maximal subgroups of various different natures in SL(n, ?). In particular, we prove that there are 2^?0 maximal subgroups, we provide a maximal subgroup whose action on the projective space has no dense orbits, and we produce a faithful primitive permutation representation of PSL(n, ?) which is not 2-transitive.     

On construction of vertex-transitive partitions of the <Emphasis Type="Italic">n</Emphasis>-cube into perfect codes

Two methods are presented to construct some vertex-transitive and 2-transitive partitions of the n-cube into perfect codes. Some lower bounds are given on the number of transitive, vertex-transitive, and 2-transitive partitions of the n-cube into perfect codes.     

Transitive affine planes admitting elations

Affine planes which admit a point transitive collineation group and at least one affine elation are considered. Such a plane is shown to be (A,?_∞)-transitive for some point A on ?_t8 and to be a translation plane if at least two distinct elation centers exist. If the plane has at least (order)^1/2+1 distinct elation centers and the group generated by the elations is nonsolvable then the plane is either Desarguesian or Lüneburg-Tits.     

Tits's theorem for the groupPGL 2 (L), L a not necessarily commutative local ring

Summary We give a characterization of the group PGL₂(L), where L is a (not necessarily commutative) local ring. So we generalize a well-know Theorem of Tits on the 3-transitive permutation groups.