Archiv der Mathematik最新文献

In this paper, the structure of uncountable groups with finitely many normalizers of large subgroups is studied and the connections between this property and other natural finiteness conditions on large subgroups of uncountable groups are investigated. In particular, groups in which every large subgroup is close to be normal with the only obstruction of a finite section and groups with finitely many commutator subgroups of large subgroups are considered. Moreover, groups with a finite covering consisting of groups with normal large subgroups are studied.

A short proof of the elliptical range theorem concerning the numerical range of (2times 2) complex matrices is given.

We provide a full classification of the slope stability of the cotangent bundles of relatively minimal smooth Weierstrass fibrations. The classification only depends on the topological Euler characteristic of the surface and the genus of the base curve.

In this paper, we are concerned with the structure of tame symmetric algebras (Lambda ) of period four (TSP4 algebras for short). For a tame algebra, the number of arrows starting or ending at a given vertex cannot be large. Here we will mostly focus on the case when the Gabriel quiver of (Lambda ) is biserial, that is, there are at most two arrows ending and at most two arrows starting at each vertex. We present a range of properties (with relatively short proofs) which must hold for the Gabriel quiver of such an algebra. In particular, we show that triangles (and squares) appear naturally, so as for weighted surface algebras (Erdmann and Skowroński in J Algebra 505:490–558, 2018, J Algebra 544:170–227, 2020, J Algebra 569:875–889, 2021). Furthermore, we prove results on the minimal relations defining the ideal I for an admissible presentation of (Lambda ) in the form KQ/I. This will be the input for the classification of all TSP4 algebras with biserial Gabriel quiver.

Abstract

In this article, we study the monogenity of a tower of number fields defined by the iterates of a stable polynomial. We give a necessary condition for the monogenity of the number fields defined by the iterates of a stable polynomial. When the stable polynomial is of certain type, we also give a sufficient condition for the monogenity of the fields defined by each of its iterate. As a consequence, we obtain an infinite 3-tower of monogenic number fields. Moreover, we construct an infinite family of stable polynomials such that each of its iterate is non-monogenic.

Abstract

We construct a weight 1/2 multiplier system for the group (Gamma _0^+(p)) , the normalizer of the congruence subgroup (Gamma _0(p)) where p is an odd prime, and we define an analogue of the eta function and Rademacher symbol and relate it to the geometry of edge paths in a triangulation of the upper half-plane.

Abstract

A group-word w is called concise if the verbal subgroup w(G) is finite whenever w takes only finitely many values in a group G. It is known that there are words that are not concise. In particular, Olshanskii gave an example of such a word, which we denote by (w_o) . The problem whether every word is concise in the class of residually finite groups remains wide open. In this note, we observe that (w_o) is concise in residually finite groups. Moreover, we show that (w_o) is strongly concise in profinite groups, that is, (w_o(G)) is finite whenever G is a profinite group in which (w_o) takes less than (2^{aleph _0}) values.

The analogues of the (L_p) mixed volumes inequality, the (L_p) Brunn–Minkowski inequality, the (L_p) Blaschke–Santaló inequality, and the (L_p) Petty projection inequality in Gaussian space are established.

Let p be an odd prime. In this article, we investigate the number of ways in which a quadratic residue and a non-residue modulo p can be expressed as sum of two quadratic residues sum of two quadratic non-residues, and sum of a quadratic residue and non-residue in an elementary way using Gauss sums.

Abstract

We describe finite groups whose principal block contains only characters of prime power degree.