On translation planes of orderq 3 that admit a collineation group of orderq 3

Let II be a translation plane of orderq ³, with kernel GF(q) forq a prime power, that admits a collineation groupG of orderq ³ in the linear translation complement. Moreover, assume thatG fixes a point at infinity, acts transitively on the remaining points at infinity andG/E is an abelian group of orderq ², whereE is the elation group ofG.In this article, we determined all such translation planes. They are (i) elusive planes of type I or II or (ii) desirable planes.Furthermore, we completely determined the translation planes of orderp ³, forp a prime, admitting a collineation groupG of orderp ³ in the translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity. They are (i) semifield planes of orderp ³ or (ii) the Sherk plane of order 27.     

p-Pseudomeasures and closed subgroups

LetPM _p (G) be the space of allp-pseudomeasures on a locally compact groupG. We show the existence of a conditional expectation fromPM _p (G) ontoPM _p (H) whereH is a closed normal subgroup ofG. As an application we give a new proof of the fact thatH is a set ofp-synthesis inG; we also get an inequality involving the operator norm of bounded measures onG. Moreover, in analogy with a theorem of Reiter, we obtain a result concerning the closed ideals of the Figà-Talamanca Herz algebra ofG.     

Collineation Groups Preserving a Unital in a Projective Plane of Order m2 with m {equiv} 1(4)

The structure of a collineation group G preserving a unitalin a projective plane of order m² with m 1(4) is investigatedand several results are obtained about G, when G is a 2-group.These results are then used to investigate the structure ofG/O(G) in the general case and, in particular, that of F*(G/O(G)).     

ON STABLE LINE SEGMENTS IN ALL TRIANGULATIONS OF A PLANAR POINT SET

Abstract. Let S be a set of finite plauar points. A llne segment L(p, q) with p, q E Sis called a stable line segment of S, if there is no Line segment with two endpoints in S intersecting L(p, q). In this paper, some geometric properties of the set of all stable line segments     

Extensions of group-valued set functions

The problem of common extension ofcharges (finitely additive measures) is generalised to include group-valued functions defined on a system of sets (u-systems). To eachu-systemU an Abelian groupH(U) is attached. Every Abelian group is isomorphic to one of the formH(U). The groupH(U) is an indicator for extendability of charges fromU to the Boolean algebra generated byU. AllG-valued measures extend if and only if Ext(H(U),G)=0, for instance. Supported as van Vleck visiting professor at Wesleyan University, Connecticut in 1993. Partially supported by the Graduierten KollogTheoretische und experimentelle Methoden der reinen Mathematik of Essen University, a project No. G-0294-081.06/93 of the German-Israeli Foundation for Scientific Research & Development and by the German Academic Exchange, DAAD 1994.     

Finite arithmetic subgroups ofGL n,III

LetG be an algebraic group inGL _n (C) defined over Q, andK an algebraic number field with the maximal orderO _k . If the groupG(O _k ) of rational points ofG inM _n (O _k ) is a finite group and if it satisfies a certain condition, which is satisfied, for example, whenK is a nilpotent extension of Q and 2 is unramified, thenG(O _k ) is generated by roots of unity inK andG(Z). Dedicated to the memory of Professor K G Ramanathan     

Linear preservers of orthogonal decomposable Tensors

Let G be a subgroup of the symmetric group S_m and V be an n-dimensional unitary space where nm. Let V(G) be the symmetry class of tensors over V associated with G and the identity character. Let D(G) be the set of all decomposable elements of V(G) and O(G) be its subset consisting of all nonzero decomposable tensors x ₁ ^?…^? x_m such that {x ₁,…,x_m } is an orthogonal set. In this paper we study the structure of linear mappings on V(G) that preserve one of the following subsets: (i)O(G), (ii) D(G)\(O(G)?{0}).     

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeL _n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L _n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.     

Some remarks on subgroups determined by certain ideals in integral group rings

LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andH∩K⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI ³(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I ²(G)I(H)+I(G)I(H)I(G)+I(H)I²(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH ²(G/H′, T)≤1, are computed. the subgroup ofG determined byI ⁿ(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained     

The poset on connected induced subgraphs of a graph need not be Sperner

SupposeG is a finite connected graph. LetC(G) denote the inclusion ordering on the connected vertex-induced subgraphs ofG. Penrice asked whetherC(G) is Sperner for general graphsG. Answering Penrice's question in the negative, we present a treeT such thatC(T) is not Sperner. We also construct a related distributive lattice that is not Sperner.     

Isoperimetric inequalities in potential theory

Given a non-empty bounded domainG in ⁿ ,n2, letr ₀(G) denote the radius of the ballG ₀ having center 0 and the same volume asG. The exterior deficiencyd _e (G) is defined byd _e (G)=r _e (G)/r ₀(G)–1 wherer _e (G) denotes the circumradius ofG. Similarlyd _i (G)=1–r _i (G)/r ₀(G) wherer _i (G) is the inradius ofG. Various isoperimetric inequalities for the capacity and the first eigenvalue ofG are shown. The main results are of the form CapG(1+cf(d _e (G)))CapG ₀ and ₁(G)(1+cf(d _i (G)))₁(G ₀),f(t)=t ³ ifn=2,f(t)=t ³/(ln 1/t) ifn=3,f(t)=t ^(n+3)/2 ifn4 (for convex G and small deficiencies ifn3).     

On a class of groups of prime-power order

LetG be a finitep-group,d(G)=dimH ¹ (G, Z _p) andr(G)=dimH ²(G, Z_p). Thend(G) is the minimal number of generators ofG, and we say thatG is a member of a classG _p of finitep-groups ifG has a presentation withd(G) generators andr(G) relations. We show that ifG is any finitep-group, thenG is the direct factor of a member ofG _p by a member ofG _p .     

On a geometric property of perfect graphs

LetG be a graph,VP(G) its vertex packing polytope and letA(G) be obtained by reflectingVP(G) in all Cartersian coordinates. Denoting byA*(G) the set obtained similarly from the fractional vertex packing polytope, we prove that the segment connecting any two non-antipodal vertices ofA(G) is contained in the surface ofA(G) and thatG is perfect if and only ifA*(G) has a similar property.     

A note on a theorem of W. Gaschütz and N. Itô

Given a finite groupG andp an odd prime number, we conclude thatO ^p(G)G isp-nilpotent when for every subgroupH ofG of orderp there exists a subgroupK ofG such thatG=HK andH permutes with every subgroup ofK.     

On automorphism groups of Cayley graphs

LetX _G,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(X_G,H) must contain the regular subgroupL _G corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(X_G,H)L_G] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.     

Structure of digraphs associated with quadratic congruences with composite moduli

We assign to each positive integer n a digraph G(n) whose set of vertices is H={0,1,…,n-1} and for which there exists a directed edge from a∈H to b∈H if . Associated with G(n) are two disjoint subdigraphs: G₁(n) and G₂(n) whose union is G(n). The vertices of G₁(n) correspond to those residues which are relatively prime to n. The structure of G₁(n) is well understood. In this paper, we investigate in detail the structure of G₂(n).     

Matching behaviour is asymptotically normal

Ak-matching in a graphG is a set ofk edges, no two of which have a vertex in common. The number of these inG is writtenp(G, k). Using an idea due to L. H. Harper, we establish a condition under which these numbers are approximately normally distributed. We show that our condition is satisfied ifn=|V(G)| is large compared to the maximum degree Δ of a vertex inG(i.e. Δ=o(n)) orG is a large complete graph. One corollary of these results is that the number of points fixed by a randomly chosen involution in the symmetric groupS is asymptotically normally distributed.     

Generalized frobenius groups. II

A pair (G, K) in whichG is a finite group andK a normal nontrivial proper subgroup ofG is said to be an F2-pair (a Frobenius type pair) if |C _G (x)|=|C _G/K (xK)| for allx∈G\K. A theorem of Camina asserts that in this case eitherK orG/K is ap-group or elseG is a Frobenius group with Frobenius kernelK. The structure ofG will be described here under certain assumptions on the Sylowp-subgroups ofG. This author’s research was partially supported by the Technion V.P.R. fund — E.L.J. Bishop research fund. This author’s research was partially supported by the MPI fund.     

A characterization ofL n(K) as a permutation group

In 1972 M. O'Nan proved thatL _n (q),h 3; can be characterized as a doubly-transitive groupG on a finite set , whereG _a has an Abelian normal subgroup acting not semi-regularly on -a. In the Main Theorem we show that a similar statement holds if is infinite. Our result implies O'Nan's theorem.This paper is part of the author's Ph.D. thesis written under supervision of Prof. F. G. Timmesfeld.     

On flag-transitive affine planes of orderq 3

Let be a translation plane of orderq ³,q an odd prime power, whose kern GF(q). Letl be the line at infinity of . LetG be a solvable collineation group of in the linear translation complement, which acts transitively onl , and letH be a maximal normal cyclic subgroup ofG. Then the restriction ofH onl acts semiregularly onl and {1, 2, 3, 6}, where is the restriction ofG onl (ifq –1(mod 3), then {1, 2}). Ifq {3, 5} and {1, 2}, then is determined completely, using a computer.