The K_2-groups over Finaite Commutative Rings

The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.

Full-Rank Factoring of Elementary 2-Groups with Equal Size Factors

In order to answer a question motivated by constructing substitution boxes in block ciphers we will exhibit an infinite family of full-rank factorizations of elementary 2-groups into two factors having equal sizes.

某些非循环2-群模表示空间的强平坦性和平坦性（英文）

研究特征为2域上,某些非循环2-群模表示空间的强平坦性和平坦性,确定对应模不变式环的深度。在阶数不为4的非循环交换2-群的不可分解表示中,有两种表示空间的类型是平坦的。对于非交换2群而言,如果群G是阶数为2^(n+2)的二面体群或广义四元数群,则其在维数为1+2~n的忠实表示都是平坦的,但都不是强平坦的。

The Isomorphism Problem of Normalized Unit Groups of Group Algebras of a Class of Finite 2-groups

Let p be a prime and F_p be a finite field of p elements.Let F_(pG)denote the group algebra of the finite p-group G over the field F_(p)and V(F_(pG))denote the group of normalized units in F_(pG).Suppose that G and H are finite p-groups given by a central extension of the form 1→Z_(p)^(m)→G→Z_(p)×···×Z_(p)→1 and G'≌Z_(p),m≥1.Then V(F_(p)G)≌V(F_(p)H)if and only if G≌H.Balogh and Bovdi only solved the isomorphism problem when p is odd.In this paper,the case p=2 is determined.

On the Structures of K2(Z[G]), G a Finite Abelian p-Group

Let G be a finite abelian p-group,Г the maximal Z-order of Z[G]. We prove that the 2-primary torsion subgroups of K2Z[G]) and K2(Г)are isomorphic when p ≡ 3, 5, 7 (mod 8, and K2(Z[G])■zZ[1/p] is isomorphic to K2(Г)■zZ[1/p] when p = 2,3,5,7. As an application, we give the structure of K2(Z[G]) for G a cyclic p-group or an elementary abelian p-group.

A Characterization of the Smallest Suzuki 2-Group

Let G be a finite nonabelian group which has no abelian maximal subgroups and satisfies that any two non-commutative elements generate a maximal subgroup. Then G is isomorphic to the smallest Suzuki 2-group of order 64.

A Classification of Finite Metahamiltonian p-Groups

A finite non-abelian group G is called metahamiltonian if every subgroup of G is either abelian or normal in G.If G is non-nilpotent,then the structure of G has been determined.If G is nilpotent,then the structure of G is determined by the structure of its Sylow subgroups.However,the classification of finite metahamiltonian p-groups is an unsolved problem.In this paper,finite metahamiltonian p-groups are completely classified up to isomorphism.

Bioinspired Noncyclic Transfer Pathway Electron Donors for Unprecedented Hydrogen Production

Electron donors are widely exploited in visible-light photocatalytic hydrogen production.As a typical electron donor pair and often the first choice for hydrogen production,the sodium sulfide-sodium sulfite pair has been extensively used.However,the resultant thiosulfate ions consume the photogenerated electrons to form an undesirable pseudocyclic electron transfer pathway during the photocatalytic process,strongly limiting the solar energy conversion efficiency.Here,we report novel and bioinspired electron donor pairs offering a noncyclic electron transfer pathway that provides more electrons without the consumption of the photogenerated electrons.Compared to the state-of-the-art electron donor pair Na_(2)S-Na_(2)SO_(3),these novel Na_(2)S-NaH_(2)PO_(2)and Na_(2)S-NaNO_(2)electron donor pairs enable an unprecedented enhancement of up to 370%and 140%for average photocatalytic H_(2)production over commercial CdS nanoparticles,and they are versatile for a large series of photocatalysts for visible-light water splitting.The discovery of these novel electron donor pairs can lead to a revolution in photocatalysis and is of great significance for industrial visible-light-driven H_(2)production.

On the torsion in K_2 of a field

For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field.

Derived 2-functors in(2-SGp)

In this paper, we construct the projective resolution of arbitrary symmetric 2-group, define thederived 2-functors in (2-SGp) and give some related properties of the derived 2-functors.

The Structure of Certain K_2O_F

In this paper, we investigate tile structure of K2OF for F = -3 mod 9 and d ≠ -3. We find the element of order 3 of K2OF for F = and generated elements of K2OF /(2) /(8) /(3) for F = . We get the property of 2F, which develops a Tate and Bass's theorem, and give the structure of K2OF for F = and the presentation relations of SLn(OF)(n ≥ 3)

MULTI-PERIODIC SOLUTIONS TO A CLASS OF SECOND-ORDER NEUTRAL DIFFERENTIAL EQUATIONS

By means of variational structure and Z 2 group index theory,we obtain infinite periodic solutions to a class of second-order neutral differential equations.