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.     

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.     

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.     

A matrix inequality on Schur complements

We investigate a matrix inequality on Schur complements defined by {1}-generalized inverses, and obtain simultaneously a necessary and sufficient condition under which the inequality turns into an equality. This extends two existing matrix inequalities on Schur complements defined respectively by inverses and Moore-Penrose generalized inverses (see Wang et. al. [Lin. Alg. Appl., 302–303 (1999) 163–172] and Liu and Wang [Lin. Alg. Appl., 293(1999) 233–241]). Moreover, the non-uniqueness of {1}-generalized inverses yields the complicatedness of the extension.     

A new characterization of Sobolev spaces

Our main result is the following: Let $g\in L^p({\mathbb{R}}^N)$, $1<p<+\infty$, be such that

$\underset{n\in \mathbb{N}}{\sup }\underset{|g(x)?g(y)|>{\delta }_n}{\underset{{\mathbb{R}}^N}{\int }\underset{{\mathbb{R}}^N}{\int }}\frac{{\delta }_n^p}{{|x?y|}^{N+p}}\mathrm{d}x\mathrm{d}y<+\infty ,$

for some arbitrary sequence of positive numbers ${({\delta }_n)}_{n\in \mathbb{N}}$ with ${\lim }_{n\to \infty }{\delta }_n=0$. Then $g\in W^{1,p}({\mathbb{R}}^N)$.This extends a result from H.-M. Nguyen (2006). To cite this article: J. Bourgain, H.-M. Nguyen, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

The saturation of frobenius powers of ideals

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