Transversal designs with λ >1 associated with Frobenius groups

We describe the transversal translation designs with gl > 1 which admit a Frobenius group as their (full) translation group.     

More balanced incomplete block designs from frobenius groups

From a Frobenius group G = N ×_σ Φ with kernel N and complement Φ, one can sometimes construct new balanced incomplete block designs by taking as blocks the translations in N of blocks made from the union of the nontrivial orbits of Φ of a and −a in N together with the trivial orbit {0}.     

Polynomials with frobenius galois groups

An effective characterization of polynomials of degree n whose Galois groups are Frobenius groups with kernel of order n is given. Some examples of such polynomials are listed.     

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.     

Relation between frobenius and 2-frobenius groups with order components of finite groups

IfG is a finite group, we define its prime graph Г(G), as follows: its vertices are the primes dividing the order ofG and two verticesp, q are joined by an edge, if there is an element inG of orderpq. We denote the set of all the connected components of the graph Г(G) by T(G)= {π_i(G), fori = 1,2, …,t(G)}, where t(G) is the number of connected components of Г(G). We also denote by π(n) the set of all primes dividingn, wheren is a natural number. Then ¦G¦ can be expressed as a product of m₁, m₂, …, m_t(G), where m_i’s are positive integers with π(m_i) = π_i. Thesem _i ’s are called the order components ofG. LetOC(G) := {m ₁,m ₂, …,m _t (G)} be the set of order components ofG. In this paper we prove that, if G is a finite group andOC(G) =OC(M), where M is a finite simple group witht(M) ≥ 2, thenG is neither Frobenius nor 2-Frobenius.     

Characterization of modular frobenius groups of special type

In this article, we first investigate the properties of modular Frobenius groups. Then, we consider the case that G' is a minimal normal subgroup of a modular Frobenius group G. We give the complete classification of G when G' as a modular Frobenius kernel has no more than four conjugacy classes in G.     

Classification of designs with nontrivial automorphism groups

In this article, we introduce a new orderly backtrack algorithm with efficient isomorph rejection for classification of t‐designs. As an application, we classify all simple 2‐(13,3,2) designs with nontrivial automorphism groups. The total number of such designs amounts to 1,897,386. The decomposability of the designs is also considered. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 479–489, 2006     

Some new designs with prescribed automorphism groups

We establish the existence of simple designs with parameters 2‐(55, 10, 4), 3‐(20, 5, 4), 3‐(21, 7, 30), 4‐(15, 5, 2), 4‐(16, 8, 45), 5‐(16, 7, 10), and 5‐(17, 8, 40), which have previously been unknown. For the corresponding t, v, and k, we study the set of all λ for which simple t‐ designs exist.     

Divisible designs and groups

We study (s, k, ₁, ₂)-translation divisible designs with ₁0 in the singular and semi-regular case. Precisely, we describe singular (s, k, ₁, ₂)-TDD's by quasi-partitions of suitable quotient groups or subgroups of their translation groups. For semi-regular (s, k, ₁, ₂)-TDD's (and, more general, for the case ₂>₁) we prove that their translation groups are either Frobenius groups or p-groups of exponent p. Some examples are given for the singular, semi-regular and regular case.     

Automorphism groups of designs with λ=1

If $G$ is a finite group and $k=q>2$ or $k=q+1$ for a prime power $q$ then, for infinitely many integers $v$, there is a 2-$(v,k,1)$-design $\mathbf{D}$ for which $\mathrm{Aut}\mathbf{D}?G$.     

The failure of the topological frobenius property for nilpotent Lie groups. II

Mathematische Annalen -     

On Hadamard designs associated with a hyperoval

Hadamard designs which can be associated with a hyperoval of a projective plane of even order are investigated. In particular, when is a translation hyperoval, these designs are shown to contain restrictions that are isomorphic to the 2-design of points and hyperplanes of a projective geometry overGF(2).     

Semi-regular divisible designs and Frobenius groups

We study and characterize semi-regular (s, k, λ₁, λ₂)-divisible designs which admit a Frobenius group as their translation group. Moreover, we give a construction method for such designs by generalized admissible triads.