The Coset Group and Com ove Group

§１．　ＢａｓｉｃＤｅｆｉｎｉｔｉｏｎｓＬｅｔＸａｎｄＹｂｅｓｅｔｓ，ｆｏｒａｇｉｖｉｎｇｍａｐｐｉｎｇｆ：Ｘ→Ｙ，ｘ｜→ｙ＝ｆ（ｘ），ａｍａｐｐｉｎｇｆｏｎｐｏｗｅｒｓｅｔｃａｎｂｅｃｏｎｄｕｃｔｅｄｂｙｆ：ｆ：Ｐ（Ｘ）→Ｐ（Ｙ），Ａ｜→Ｂ＝ｆ（Ａ）＝｛ｙ｜ｘ∈Ａ，ｙ＝ｆ（ｘ）｝．　　Ｎｏｔｅｔｈａｔａｂｉｎａｒｙｏｐｅｒａｔｉｏｎｉｓａｓｐｅｃｉａｌｍａｐｐｉｎｇｏｎｓｅｔｓ．ＧｉｖｅｎａｂｉｎａｒｙｏｐｅｒａｔｉｏｎｏｎｓｅｔＸｓｕｃｈｔｈａｔ：：Ｘ×Ｘ→Ｘ，　（ｘ，ｙ）｜→ｘ…     

幂群所诱导的L－Fuzzy幂群

本文在幂群［１］的基础上，提出了幂群所诱导的Ｌ－Ｆｕｚｚｙ幂群、Ｌ－Ｆｕｚｚｙ幂群的λ－截、次幂群等—系列概念，并给出了幂群与其诱导的Ｌ－Ｆｕｚｚｙ幂群、λ截集群间的次同态、同态等关系。     

一类交换群的自同构群

本文利用一种新方法,考虑交换群Cp2×Cp 的自同构群,得到了该自同构群的结构.我们的结论是:Aut(Cp2×Cp)[Cp- 1∝(Cp×Cp)]holCp.     

格群的T—Fuzzy子格群

引入格群L关于t-norm T的Fuzzy子格群、Fuzzy正则子格群（简称T-Fuzzy子格群、T-Fuzzy正规子格群）等概念,研究它们的性质,给出T-Fuzzy子格群的刻画。     

直觉模糊群的子直觉模糊群和正规子直觉模糊群

设G为直觉模糊二元运算下的直觉模糊群,给出了G子直觉模糊群和正规子直觉模糊群的定义,讨论并证明了子直觉模糊群和正规子直觉模糊群的一些性质。     

自同构群

设Ｇ是有限群，ＡｕｔＧ＝ＡＢ，Ｂ△ＡｕｔＧ，（｜Ａ｜，｜Ｂ｜）＝１，Ａ是交换群且每Ｓｙｏｗ子群的秩≤２，Ｂ是循环群。本文得出了Ｇ的结构，特别地，证明了ＡｕｔＧ是秩≤的交换群时，Ｇ循环。     

考虑群体一致性的动态群体决策方法

为了求解一类包含多轮群体评价过程的动态群体决策问题，定义了个体效用波动和群体一致度的概念并分别建立了相应的计算指标，利用决策个体的效用波动指标提出了决策个体权重的修正方法，然后提出了一种基于群体一致度指标的加权算法，得到了各决策方案的群体效用评价。最后给出了计算实例。     

自同构群是循环群被交换群扩张的有限群

设Ｃ是有限群，ＡｕｔＧ＝ＡＢ，，Ａ是交换群且每Ｓｙｌｏｗ子群的秩≤２，Ｂ是循环群，本文得出了Ｇ的结构，特别地，证明了ＡｕｔＧ是秩≤２的交换群时，Ｇ循环。     

The Group of Automorphisms of the Rational Group Algebra of a Finite Metacyclic Group

We investigate the group of automorphisms Aut(? G) of a rational group algebra ? G of a finite metacyclic group G by first describing the simple components of the Wedderburn decomposition of ? G and then investigating when two of these simple components are isomorphic.     

The Lie Group Structure of the Butcher Group

The Butcher group is a powerful tool to analyse integration methods for ordinary differential equations, in particular Runge–Kutta methods. In the present paper, we complement the algebraic treatment of the Butcher group with a natural infinite-dimensional Lie group structure. This structure turns the Butcher group into a real analytic Baker–Campbell–Hausdorff Lie group modelled on a Fréchet space. In addition, the Butcher group is a regular Lie group in the sense of Milnor and contains the subgroup of symplectic tree maps as a closed Lie subgroup. Finally, we also compute the Lie algebra of the Butcher group and discuss its relation to the Lie algebra associated with the Butcher group by Connes and Kreimer.     

Group Entwining Structures and Group Coalgebra Galois Extensions

Abstract

The notions of group coalgebra Galois extension and group entwining structure are defined. It is proved that any group coalgebra Galois extension induces a unique group-entwining map ψ = {ψ_{α, β}}_{α, β∈π} compatible with the right group coaction, generalizing the recent work of Brzeziński and Hajac [Brzeziński, T., Hajac, P. M. (1999). Coalgebra extensions and algebra coextensions of Galois type. Comm. Algebra 27:1347–1368].     

Equivariant Mapping Class Group and Orbit Braid Group

Motivated by the work of Birman about the relationship between mapping class groups and braid groups, the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper. Their construction is based on the exact sequence given by the fibration F₀^GM → F(M/G, n). The conclusion is closely connected with the braid group of the quotient space. Comparing with the situation without the group action, there is a big difference when the quotient space is T².     

Group C-Rings and Weak Group Algebra Galois Coextensions

Abstract

We use the classification of finite order automorphisms by Kac to characterize all maximal subalgebras, regular, semisimple, reductive or not of a simple complex Lie algebra (up to conjugacy) that we can determine from its Dynkin diagram. Using Barnea et al. [Barnea, Y., Shalev, A., Zelmanov, E. I. (1998). Graded subalgebras of affine Kac–Moody algebras. Israel J. Math. 104:321–334] we extend our results to the case of affine Kac–Moody algebras. We also point out some inaccuracies in the Dynkin paper [Dynkin, E. B. (1957a). Semisimple subalgebras of semisimple Lie algebras. Amer. Math. Soc. Transl t. 6:111–244].     

Group rings

It is proved that from the isomorphism of the integral group rings of two groups follows the isomorphism of the quotient-groups of these groups by the second commutator subgroup. Groups with certain conditions T₁ and T₂ are considered. From condition T₁ it follows that the group algebra over an integral domain has no zero-divisors, and from T₂ that the group of invertible elements of this algebra is trivial. Sufficient conditions are given, under which there are no zero-divisors in the group algebra of a pro-p-group over a field of p elements.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 140–145, 1980.     

Group Corings

We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the G-comodules over a Galois group coring is given. We study (graded) Morita contexts associated to a group coring. Our theory is applied to group corings associated to a comodule algebra over a Hopf group coalgebra. This research was supported by the research project G.0622.06 “Deformation quantization methods for algebras and categories with applications to quantum mechanics” from Fonds Wetenschappelijk Onderzoek-Vlaanderen. The third author was partially supported by the SRF (20060286006) and the FNS (10571026).