Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable

Thompson’s theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson’s theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.     

Directoid groups

We continue the study of directoid groups, directed abelian groups equipped with an extra binary operation which assigns an upper bound to each ordered pair subject to some natural restrictions. The class of all such structures can to some extent be viewed as an equationally defined substitute for the class of (2-torsion-free) directed abelian groups. We explore the relationship between the two associated categories, and some aspects of ideals of directoid groups.     

Functions realizing as abelian group automorphisms

Let A be a set and f:A→A a bijective function. Necessary and sufficient conditions on f are determined which makes it possible to endow A with a binary operation ? such that (A,?) is a cyclic group and f∈Aut(A). This result is extended to all abelian groups in case |A| = p², p a prime. Finally, in case A is countably infinite, those f for which it is possible to turn A into a group (A,?) isomorphic to ?ⁿ for some n≥1, and with f∈Aut(A), are completely characterized.     

Difference Sets with n = 2pm

Let D be a (v,k,) difference set over an abelian group G with even n = k - . Assume that t N satisfies the congruences t q _i ^fi (mod exp(G)) for each prime divisor q_i of n/2 and some integer f_i. In [4] it was shown that t is a multiplier of D provided that n > , (n/2, ) = 1 and (n/2, v) = 1. In this paper we show that the condition n > may be removed. As a corollary we obtain that in the case of n= 2p^a when p is a prime, p should be a multiplier of D. This answers an open question mentioned in [2].     

A note on finite abelian covers

We prove that any abelian cover over a smooth variety is defined by some cyclic equations. From the defining equations, we compute explicitly the normalization, branch locus, ramification indices, global invariants, and the resolution of singularities. As an application, we construct a new algebraic surface which is the quotient of ball.     

几何代数基础新视角下的初步探讨

提出有关几何代数基础的一个问题:在给定了变换群的几何上,可能建立哪些代数结构?首先证明,不可能在欧氏平面上的点之间定义一种在保距变换下不变的运算,使之在此运算下形成阿贝尔群.进一步的讨论证明,只有将欧氏几何扩大为质点几何,才能在其上建立在保距变换下不变的可交换可结合的运算,而且这种运算只能是质点几何中的加法.如果希望在此运算下构成阿贝尔群,就必须引入向量.最后讨论了所获结果的意义,并提出若干问题.     

最大Abel商群为局部循环群的可解群

从自同构群的角度出发,给出了一些具有有限性条件的、最大Abel商群为局部循环群的可解群的结构．     

On Separable Schur Rings over Abelian Groups

A finite group is said to be weakly separable if every algebraic isomorphism between two S-rings over this group is induced by a combinatorial isomorphism.We prove that every abelian weakly separable group only belongs to one of several explicitly given families.     

Isomorphisms of Cayley graphs of a free Abelian group

A group G is called a CI-group provided that the existence of some automorphism σ ∈ Aut(G) such that σ(A) = B follows from an isomorphism Cay(G, A) ? = Cay (G, B) between Cayley graphs, where A and B are two systems of generators for G. We prove that every finitely generated abelian group is a CI-group.     

Zero-sum problems in finite abelian groups: A survey

We give an overview of zero-sum theory in finite abelian groups, a subfield of additive group theory and combinatorial number theory. In doing so we concentrate on the algebraic part of the theory and on the development since the appearance of the survey article by Y. Caro in 1996.     

所有子群皆交换或正规的有限群

本文研究所有子群皆交换或正规的有限群. 我们获得了非幂零的情形的一个特征刻画, 也给出了幂零情形的一些性质.     

Automorphism Groups of Finite Abelian p-groups and Transvections

We define a particular type of automorphisms called transvections on a finite finite abelian p-group H_p. It is proved that the subgroup E of the automorphism group Aut(H_p) of H_p generated by those transvections is normal in it, and that Aut(H_p) can be written as the product of E and some abelian subgroup K. The center of Aut(H_p) is also determined.     

关于有限Abel p-群的自同构群

从有限Abel p-群P的型不变量出发,给出了其自同构群AutP的阶的计算公式,并利用|AutP|的计算公式得到了下面3个结果:1.由有限Abel p-群的型不变量的两种变换得到了其自同构群的阶的变化规律;2.用群的阶、秩、幂指数三个量界定了有限Abel p-群的自同构的阶;3.对部分Frattini子群为p阶群的有限p-群,确定了其自同构群的阶何时达到最小值和最大值.     

On birational transformations of pairs in the complex plane

This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of the art and provide some new results on this subject.      

Serre''s contribution to the development of algebraic topology

We describe in detail Serre's application of spectral sequence theory to the study of the relations between the homology of total space, base space and fibre in a Serre fibration; and we apply the results to establish that a 1-connected space X has homology groups (in positive dimension) in a Serre class C if and only if its homotopy groups are in C.

We include in this paper some personal reflections on the contact the author had with Serre during the decade of the 1950's when Serre's revolutionary work in homotopy theory was completely changing the face of algebraic topology.     

A Problem on Zero Subsums in Abelian Groups

Let a₁, a₂, ... be elements of an abelian group such that a_mhas order larger than m^m. Then the multiset of the a_m's can be partitioned into two parts which are free of zero subsums. The result was motivated by a question of Pál Erds.     

Hereditary triangular matrix comonads

We recognize Harada’s generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with comodules over comonads. With this conceptual tool at hand, we obtain several of the Harada results with simpler proofs, some of them under more general hypothesis, besides with a characterization of the normal triangular matrix comonads that are hereditary, that is, of homological dimension less than or equal to 1. Our methods rest on a matrix representation of additive functors and natural transformations, which allows us to adapt typical algebraic manipulations from Linear Algebra to the additive categorical setting.     

A characterization of alternating groups by the set of orders of maximal abelian subgroups

We prove that alternating groups with three prime graph components are uniquely determined by the set of orders of maximal abelian subgroups.     

A Factorization Theorem for Topological Abelian Groups

Using the nice properties of the w-divisible weight and the w-divisible groups, we prove a factorization theorem for compact abelian groups K; namely, K = K _tor  × K _d , where K _tor is a bounded torsion compact abelian group and K _d is a w-divisible compact abelian group. By Pontryagin duality this result is equivalent to the same factorization for discrete abelian groups proved in [9 Galindo , J. , Macario , S. ( 2011 ). Pseudocompact group topologies with no infinite compact subsets . J. Pure and Appl. Algebra 215 : 655 – 663 .[Crossref], [Web of Science ®] , [Google Scholar]].