Preprints (rote Reihe) des Fachbereich Mathematik
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Faculty / Organisational entity
208
In this paper we continue the study of p - groups G of square order \(p^{2n}\) and investigate the existence of partial congruence partitions (sets of mutually disjoint subgroups of order \(p^n\)) in G. Partial congruence partitions are used to construct translation nets and partial difference sets, two objects studied extensively in finite geometries and combinatorics. We prove that the maximal number of mutually disjoint subgroups of order \(p^n\) in a group G of order \(p^{2n}\) cannot be more than \((p^{n-1}-1)(p-1)^{-1}\) provided that \(n\ge4\)and that G is not elementary abelian. This improves a result in [6] and as we do not distinguish the cases p=2 and p odd in the present paper, we also have a generalization of D. FROHARDT' s theorem on 2 - groups in [4]. Furthermore we study groups of order \(p^6\). We can show that for each odd prime number, there exist exactly four nonisomorphic groups which contain at least p+2 mutually disjoint subgroups of order \(p^3\). Again, as we do not distinguish between the even and the odd case in advance, we in particular obtain
D. GLUCK' s and A. P. SPRAGUE' s classification of groups of order 64 which contain at least 4 mutually disjoint subgroups of order 8 in [5] and [13] respectively.
206
In this paper the existence of translation transversal designs which is equivalent to the existence of certain particular partitions in finite groups is studied. All considerations are based on the fact that the particular component of such a partition (the component representing the point classes of the corresponding design) is a normal subgroup of the translation group. With regard to groups admitting an (s,k,\(\lambda\))-partiton, on one hand the already known families of such groups are determined without using R. BAER's, 0.H.KEGEL's and M. SUZUKI' s classification of finite groups with partition and on the other hand some new results on the special structure of p - groups are proved. Furthermore, the existence of a series of nonabelian p - groups of odd order which can be represented as translation groups of certain (s,k,1) - translation transversal designs is shown; moreover, the translation groups are normal subgroups of collineation groups acting regularly on the set of flags of the same designs.
207
Moduli for singularities
(1991)
The aim of this article is to give a survey on recent results about moduli spaces for curve singularities and for modules over the local ring of a fixed curve singularity. We emphasize especially the general concept which lies behind these constructions.
Therefore, the article might be useful to the reader who wishes to have the leading ideas and the main steps of the proofs explained without going into all the details. We also calculate explicit examples (for singularities and for modules) which illustrate
the general theorems.
200
Das sind die Texte der Vorlesungen, die ich im Dezember 1988 - März 1989 an der Universität Kaiserslautern hielt. Die Sektionen 1-4 enthalten Materialien, die in Russisch im Buch [33] und in früheren Arbeiten [27,28] [30-33] publiziert sind.
Sektion 5 enthält neue Ergebnisse, die wir während meines Aufenthaltes in Kaiserslautern in Zusammenarbeit mit Herrn Robert Plato
(TU Berlin) ausarbeiteten (siehe [21,22]). Sektion 6 ist eine Erweiterung der Arbeit [31].
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The notion of Q-Gorenstein smoothings has been introduced by Kollar. ([KoJ], 6.2.3). This notion is essential for formulating Kollar's conjectures on smoothing components for rational surface singularities. He conjectures, loosely speaking, that every smoothing of a rational surface singularity can be obtained by blowing down a deformation of a partial resolution, this partial resolution having the property (among others) that the singularities occuring on it all have qG-smoothings. (For more details and precise statements see [Ko], ch. 6.). It is therefore of interest to construct singularities having qG-smoothings.
220
Hyperidentities
(1992)
The concept of a free algebra plays an essential role in universal algebra and in computer science. Manipulation of terms, calculations and the derivation of identities are performed in free algebras. Word problems, normal forms, system of reductions, unification and finite bases of identities are topics in algebra and logic as well as in computer science. A very fruitful point of view is to consider structural properties of free algebras. A.I. Malcev initiated a thorough research of the congruences of free algebras. Henceforth congruence permutable, congruence distributive and congruence modular varieties are
intensively studied. A lot of Malcev type theorems are connected to the congruence lattice of free algebras. Here we consider free algebras as semigroups of compositions of terms and more specific as clones of terms. The properties of these semigroups and clones are adequately described by hyperidentities. Naturally a lot of theorems of "semigroup" or "clone" type can be derived. This topic of research is still in its beginning and therefore a lot öf concepts and results cannot be presented in a final and polished form. Furthermore a lot of problems and questions are open which are of importance for the further development of the theory of hyperidentities.
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We present a generalization of Proth's theorem for testing certain large integers for primality. The use of Gauß sums leads to a much simpler approach to these primality criteria as compared to the earlier tests. The running time of the algorithms is bounded by a polynomial in the length of the input string. The applicability of our algorithms is linked to certain diophantine approximations of \(l\)-adic roots of unity.
219
A Remark on Primes of the Form \(2^{3n}a + 2^{2n}b+2^nc+1\). Necessary and sufficient conditions for the numbers in the title to be prime are given. The tests are well suited for practical purposes.