Hans Kurzweil

Reduced-complexity collaborative decoding of interleaved Reed-Solomon and Gabidulin codes

An alternative method for collaborative decoding of interleaved Reed-Solomon codes as well as Gabidulin codes for the case of high interleaving degree is proposed. As an example of application, simulation results are presented for a concatenated coding scheme using polar codes as inner codes.

A combinatorial technique for simplicial complexes and some applications to finite groups

In several recent papers on simplicial complexes defined on the subgroup lattice of a finite group, a central role is played by the topological notion of contractibility and, more generally homotopy equivalence of a subcomplex with the whole complex. We propose to show that these can be advantageously replaced by the following non-topological properties of a complex 9. (a) !Q is strongly collapsible (b) 9 strongly collapses to a subcomplex of R. Property (a) (termed “non-evasiveness” in [5]) has an interesting combinatorial interpretation arising from complexity questions in graph theory; its usefulness was pointed out in [5] by J. Kahn, M. Saks and D. Sturtevant. In Section 1 we collect the basic definitions and results. The central part of the paper is Section 2, where we study order complexes (defined over posets). Among other, we prove and generalize, in a rather elementary way, results obtained by G.C. Rota, A. Bjijrner and J.W. Walker in [8,1, lo]. In the remaining we apply the results of Section 2 to various complexes arising from finite groups.

A lattice theoretic characterization of prefrattini subgroups

Consider an interval [H,G] in the lattice of subgroups of a finite soluble groupG. We define a certain set of subgroups in the lattice [H,G], and prove that they are conjugate inG. ForH=1 one gets the prefrattini subgroups ofG.

Transfer and p-Factor Groups

To search for nontrivial proper normal subgroups is often the first step in the investigation of a finite group. For example, if the group G has such a normal subgroup N, then in proofs by induction one frequently gets information about N and G/N, allowing one to derive the desired result for G (e.g., 6.1.2 on page 122).

Action and Conjugation

The notion of an action plays an important role in the theory of finite groups. The first section of this chapter introduces the basic ideas and results concerning group actions. In the other two sections the action on cosets is used to prove important theorems of Sylow, Schur-Zassenhaus and Gaschutz.

Groups Acting on Groups

The action of a group A on a set G is described by a homomorphism

p-Groups and Nilpotent Groups

\pi :A \to {S_{G}}

Operieren und Konjugieren

; see Section 3.1. Suppose that G is not only a set but also a group. Then Aut G ≤ S G , and we say that π describes the action of A on the group G if Im π is a subgroup of Aut G. In other words, in this case the action of A on G not only satisfies O 1 and O 2 but also