The Isomorphism Problem for Integral Group Rings of a Complete Monomial Group

Let G be a complete monomial group with abelian base, namely, G = AwrSym _m , the wreath product of a finite abelian group A with the symmetric group on m letters. Then the group G is determined by its integral group ring.     

Zero-Divisor Graphs for Group Rings

Let R be a commutative ring with 1 ≠ 0, G be a nontrivial finite group, and let Z(R) be the set of zero divisors of R. The zero-divisor graph of R is defined as the graph Γ(R) whose vertex set is Z(R)* = Z(R)?{0} and two distinct vertices a and b are adjacent if and only if ab = 0. In this paper, we investigate the interplay between the ring-theoretic properties of group rings RG and the graph-theoretic properties of Γ(RG). We characterize finite commutative group rings RG for which either diam(Γ(RG)) ≤2 or gr(Γ(RG)) ≥4. Also, we investigate the isomorphism problem for zero-divisor graphs of group rings. First, we show that the rank and the cardinality of a finite abelian p-group are determined by the zero-divisor graph of its modular group ring. With the notion of zero-divisor graphs extended to noncommutative rings, it is also shown that two finite semisimple group rings are isomorphic if and only if their zero-divisor graphs are isomorphic. Finally, we show that finite noncommutative reversible group rings are determined by their zero-divisor graphs.     

On Homogeneous Standard Integral Table Algebras of Degree 4

In this paper we classify, up to exact isomorphism, the algebras in the title which contain a faithful element and in which every non-trivial table subset has dimension at least five.     

A Construction of Loops which admit Right Alternative Loop Rings

In this note, the authors offer a specific construction of loops whose loop rings are right, but not left, alternative.     

Linear Groups Over Integral Extensions of Semilocal Commutative Rings

Let K be an associative and commutative ring with 1, k a subring of K such that 1 ∈ k, K is an integral finitely generated extension of k, the element 2 invertible in k, and k is semilocal. The paper studies subgroups of the general linear group GL _n (K) with n ≥ 2 containing the special linear group SL _n (k).     

量子广义Kac-Moody代数的整形式

主要讨论一阶量子广义Kac-Moody代数U_q(2α)的结构,其中a∈Z0.在此基础上,刻画了量子广义代数U_q(g)的另一种整形式.     

A Note on Constants of Integral Derivations in Semiprime Rings

Let R be a semiprime ring with extended centroid C and with Q its Martindale symmetric ring of quotients. Suppose that δ: R → R is a C-integral derivation. For a subring A of Q, let A^(δ) denote the subring of constants of δ in A. We prove that R^(δ) and Q^(δ) satisfy the same polynomial identities with coefficients in C. In particular, R^(δ) is not nil of bounded index.     

Central Units in Metacyclic Integral Group Rings

In this article, we give a method to compute the rank of the subgroup of central units of ? G, for a finite metacyclic group, G, by means of ?-classes and ?-classes. Then we construct a multiplicatively independent set  ? (U(? C _{p, q} )) and by applying our results, we prove that  generates a subgroup of finite index.     

Structures of Generalized Loop Virasoro Algebras

In this article, we study the structure theory of a class of generalized loop Virasoro algebras. In particular, the derivation algebras, the automorphism groups, and the second cohomology groups of generalized loop centerless Virasoro algebras are determined.     

Unit Groups of Integral Finite Group Rings with No Noncyclic Abelian Finite p-Subgroups

It is shown that in the units of augmentation one of an integral group ring ? G of a finite group G, a noncyclic subgroup of order p ², for some odd prime p, exists only if such a subgroup exists in G. The corresponding statement for p = 2 holds by the Brauer–Suzuki theorem, as recently observed by Kimmerle.     

Homological Algebra for Coalgebraic Modules and Mod p K-Theory of Infinite Loop Spaces

In this paper, we develop the theory of homological algebra for coalgebraic modules, and compute and p K-theory of infinite loop spaces associated to BP-module spectra.     

On Noncommutative Integral Standard Generalized Table Algebras of Dimension 6

We will investigate the structure of noncommutative imprimitive association schemes of rank 6 and the representation of their adjacency algebras. From this investigation, we will construct new integral standard generalized table algebras with three parameters. These algebras are noncommutative, imprimitive, and 6-dimensional.     

The Subgroup Normalizer Problem for Integral Group Rings of Some Nilpotent and Metacyclic Groups

For a group G and a subgroup H of G, this article discusses the normalizer of H in the units of a group ring RG. We prove that H is only normalized by the “obvious” units, namely products of elements of G normalizing H and units of RG centralizing H, provided H is cyclic. Moreover, we show that the normalizers of all subgroups of certain nilpotent and metacyclic groups in the corresponding group rings are as small as possible. These classes contain all dihedral groups, all finite nilpotent groups, and all finite groups with all Sylow subgroups being cyclic.     

The Normalizer Property for Integral Group Rings of Some Finite Nilpotent-by-Nilpotent Groups

In this article, it is shown that the normalizer property holds for the following two kinds of finite nilpotent-by-nilpotent groups: (1) G = NwrH is the standard wreath product of N by H, where N is a finite nilpotent group and H is a finite abelian 2-group; (2) G is a finite group having a normal nilpotent subgroup N such that the integral group ring ?(G/N) has only trivial units. Our results generalize a result of Yuanlin Li and extend some ones obtained by Juriaans, Miranda, and Robério.     

Isomorphism of Modular Group Algebras of p-Mixed Abelian Groups

Let A be a torsion-free abelian group and F a free subgroup of A. We prove that if A/F is a reduced p-group and A/(F + C) is reduced for every p-pure subgroup C of A, then A is free.

Let KG be the group algebra of an abelian group G over a field K of prime characteristic p. Denote by S(KG) the p-component of the group V(KG) of normalized units of KG (of augmentation 1). Let H be an arbitrary group and KH ? KG as K-algebras. We prove the following. First, assume that G is a splitting group, the p-component G _p of G is simply presented, and the field K is perfect. Then H _p  ? G _p . If, in addition, G is p-mixed, then G _p is a direct factor of S(KG), and G is a direct factor of V(KG), each with the same simply presented complement. Secondly, we introduce a class of special p-mixed abelian groups and prove that, if G belong to this class, then any group basis of the group algebra KG splits. Besides, H is p-mixed and splits. Thirdly, if G is a special p-mixed abelian group and G _p is a reduced totally projective p-group, then H ? G. These results correct some essential inaccuracies and incompleteness in the proofs of results in this direction of Danchev [3-8 Danchev , P. V. ( 1998 ). Isomorphism of commutative group algebras of mixed splitting groups . Compt. Rend. Acad. Bulg. Sci. 51 : 13 – 16 . Danchev , P. V. ( 2000 ). Isomorphism of modular group algebras of totally projective abelian groups . Communications in Algebra 28 : 2521 – 2531 . Danchev , P. V. ( 2001 ). On a question of W. L. May concerning the isomorphism of modular group algebras . Communications in Algebra 29 : 1953 – 1958 . Danchev , P. V. ( 2001 ). Normed units in Abelian group rings . Glasg. Math. J. 43 : 365 – 373 . Danchev , P. V. ( 2002 ). Invariants for group algebras of splitting abelian groups with simply presented components . Compt. Rend. Acad. Bulg. Sci. 55 : 5 – 8 . Danchev , P. V. ( 2004 ). A note on the isomorphic modular group algebras of abelian groups with simply presented p-components . Compt. Rend. Acad. Bulg. Sci. 57 : 13 – 14 . ].     

The Normalizer Property for Integral Group Rings of a Class of Complete Monomial Groups

Let G = H?S _m be the natural wreath product of H by S _m , where H is a finite 2-closed group and S _m is the symmetric group of degree m. It is shown that the normalizer property holds for G.     

一类反对称loop代数及其应用

构造了一类3×3的反对称loop代数,由此设计了一个含5个位势函数的等谱问题;利用屠格式导出了一个Liouville可积系统,且拥有双Hamilton结构.     

Half-isomorphisms of Finite Automorphic Moufang Loops

We show that each half-automorphism of a finite automorphic Moufang loop is trivial. In general, this is not true for finite left automorphic Moufang loops and for finite automorphic loops.     

共振条件下二阶积分边值问题解的存在性（英文）

利用上下解方法和带参数的紧向量场解集的连通性质研究了共振条件下一类二阶微分方程积分边值问题{u′′（t）=f（t,u（t））,t∈（0,1）,u（0）=∫10u（s）dα（s）,u（1）=∫10u（s）dβ（s）解的存在性.