On the frobenius group

It is proved that forl > 1 the permutational rank of a Frobenius group with kernel of nilpotency class 1 is greater than 4^i–1+1. Let p and q be prime numbers, q¦p – 1, 1 < j < q. For all such p, q, j one constructs Frobenius groups of order p^{j + 1}q with kernel of nilpotency class j and order p^j+1. For p=7, q=3, j =2 we obtain O. Yu. Shmidt's known example.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1021–1030, July–August, 1991.     

Hopf frobenius extensions of algebras

Let AlB be an H—extension for a finite Hopf algebra H. First we characterize such Hopf extensions that are Frobenius extensions. Second we will give some characterizations of Hopf Galois extensions, which extend the result of Cohen-Fishmann-Montgomery     

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.     

On dualization of double frobenius algebras

Double Frobenius algebras (or dF-algebras) were recently introduced by the author. The concept generalizes finite-dimensional Hopf algebras, adjacency algebras of (non-commutative) association schemes, and C-algebras (or character algebras). In this paper we define a dualization construction of a dF-algebra, the so-called linear dual. We show that in the case of a Hopf algebra the linear dual gives the usual dual Hopf algebra and in the case of a C-algebra it essentially gives the usual Kawada’s dual.     

A new characterization of A 5

Let G be a group and τ _e (G) the set of numbers of elements of G of the same order. In this paper, by τ _e (G), we give a new characterization of A ₅, where A ₅ is the alternating group of degree 5. We get the theorem following: Theorem. Let G be a group, ${G\cong A_5}$ if and only if τ _e (G) = τ _e (A ₅) = {1, 15, 20, 24}.     

A new characterization of arithmeticity

We prove that an equivalence lattice is arithmetical whenever for every and there exists a compatible choice function modulo , having a as a fixed point. The converse holds if L is finite. Received September 30, 1998; accepted in final form December 1, 1998.     

Morphisms of double frobenius algebras. i

Double Frobenius algebras (or dF-algebras) were recently introduced by the author. The concept includes finite-dimensional Hopf algebras, adjacency algebras of (non-commutative) association schemes, and C-algebras (character algebras). In this paper we define certain kinds of morphisms of dF-algebras and develop their basic theory. The morphisms generalize homomorphisms of Hopf algebras and of C-algebras.     

Embedding covers and the theory of frobenius fields

We show that the theory of Frobenius fields is decidable. This is conjectured in [4], [8] and [13], and we prove it by solving a group theoretic problem to which this question is reduced there. To do this we present and develop the notion of embedding covers of finite and pro-finite groups. We also answer two other problems from [8], again by solving a corresponding group theoretic problem: A finite extension of a Frobenius field need not be Frobenius and there are PAC fields which are not Frobenius fields. Portions of this work will be incorporated in the doctoral dissertation of the first author done in Tel Aviv University under the supervision of Prof. Moshe Jarden.     

The saturation of frobenius powers of ideals

Some special cases are proved in which localization commutes with tight closure.     

On Shellsort and the frobenius problem

A boundO(N ^1+1/k ) for the running time of Shellsort, withO(logN) passes, is proved very simply by application of a Frobenius basis withk elements.     

Transversal designs associated with frobenius groups

To any Frobenius group G (of degree s, with Frobenius complement of order k) we associate an (s,k) -transversal design (G) which admits G as a point-regular collineation group. (G) is in fact also a dual translation net and furthermore admits a flag-regular collineation group. Also, (G) has two orthogonal resolutions. Conversely, we will characterize the Frobenius groups among the point-regular collineation groups of resolvable transversal designs. We also exhibit two further classes of flag-regular transversal designs. Finally, we completely determine the possible parameters of TD's constructed as above.Dedicated to Professor R. Artzy on the occasion of his 70th birthday