The minimum depth d B,A of a subring B ⊆ A introduced in the work of Boltje, Danz and Külshammer 2011 is studied and compared with the tower depth of a Frobenius extension. We show that d B,A < ∞ if A is a finite-dimensional algebra and B has finite representation type. Some conditions in terms of depth and QF property are given that ensure that the modular function of a Hopf algebra restricts ...

We give further results for Perron-Frobenius theory on the numericalrange of real matrices and some other results generalized from nonnegative matricesto real matrices. We indicate two techniques for establishing the main theorem ofPerron and Frobenius on the numerical range. In the rst method, we use acorresponding version of Wielandt's lemma. The second technique involves graphtheory.

It is over a century since Frobenius initiated the study of group representations. One feature of the subject in his day was the interplay between the general theory and the study of the important special groups (for Frobenius, the symmetric groups and PSL(2, p) for example) and this connection has continued to be a central theme of the field ever since, with ideas, questions and motivation flo...

The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each non-negative integer matrix P and positive ...

Suppose that a finite group G admits a Frobenius group of automorphisms FH with kernel F and complement H such that the fixed-point subgroup of F is trivial: CG(F ) = 1. In this situation various properties of G are shown to be close to the corresponding properties of CG(H). By using Clifford’s theorem it is proved that the order |G| is bounded in terms of |H| and |CG(H)|, the rank of G is boun...

In this paper we provide characterizations of p-nilpotency for fusion systems and p-local finite groups that are inspired by known result for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate.

This is a survey article on recent progress concerning transcendence problems over function fields in positive characteristic. We are interested in some special values that occur in the following two ways. One is the special values of certain special transcendental functions, e.g., Carlitz ζ-values at positive integers, which are specialization of Goss’ two-variable ζ-function, arithmetic (resp...