On Groups with Finite Abelian Section Rank Factorized by Mutually Permutable Subgroups

It is proved that if G = AB is a soluble group with finite abelian section rank which is factorized by two mutually permutable finite-by-nilpotent subgroups A and B such that A′ and B′ are locally nilpotent, then also the normal closure ? A′, B′ ?^G is locally nilpotent and the subgroups A′ and B′ are ascendant in G.     

非极大交换子群皆正规的有限群

设H是有限群G的一个交换子群.如果H在G中的中心化子正是它本身,则称H为G的极大交换子群.本文主要研究每一非极大交换子群都正规的有限群的结构,对这类有限群给出其完全分类.     

On Counting Certain Abelian Varieties Over Finite Fields

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers[Doc. Math., 21, 1607-1643 (2016)],[Taiwanese J. Math., 20(4), 723-741 (2016)], etc., the current authors and T. C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number q. This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the previous work by the second named author[Forum Math., 22(3), 565-582 (2010)] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.     

Uniqueness of uniform decompositions in exact categories

Two uniqueness theorems on uniform decompositions due to Krause, Diracca and Facchini are extended from abelian categories to weakly idempotent complete exact categories. We give applications to (quasi-)abelian categories, finitely accessible additive categories and exactly definable additive categories.     

On splice quotients of the form {zn = f(x,y)}

The splice quotients, defined by W. D. Neumann and J. Wahl, are an interesting class of normal surface singularities with rational homology sphere links. In general, it is difficult to determine whether or not a singularity is analytically isomorphic to a splice quotient, although there are certain necessary topological conditions. Let {zⁿ = f(x, y)} define a surface X_{f, n} with an isolated singularity at the origin in $\mathbb {C}^3$. We show that for irreducible f, if (X_{f, n}, 0) satisfies the necessary topological conditions, then there exists a splice quotient of the form (X_{g, n}, 0), where the plane curve singularity defined by g = 0 has the same topological type as the one defined by f = 0. We also present an example of an (X_{f, n}, 0) that is not a splice quotient, but for which the universal abelian cover is a complete intersection of splice type together with a non‐diagonal action of the discriminant group. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

内交换p群的中心扩张(Ⅳ)

设N,H是任意的群.若存在群G,它具有正规子群≤Z(G),使得≌N且G/≌H,则称群G为N被H的中心扩张.本文完全分类了当N为p~3阶初等交换p群及H为内交换p群时,N被H的中心扩张得到的所有不同构的群.从而我们完全分类了初等交换p群被内交换p群的中心扩张得到的所有不同构的群.     

On the tropical Torelli map

We construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map.