Dynamical Properties of Additive Cellular Automata over Finite Abelian Groups

Some dynamical properties were discussed for additive cellular automata(CA)over finite abelian groups.These properties include surjection,ergodicity,sensitivity to initial conditions and positive expansivity.Some necessary and sufficient conditions of determining ergodicity and sensitivity of the above additive CA were presented,respectively.A necessary condition for the positive expansivity of the above additive CA was given.The positive expansivity was proved to be preserved under the shift mappings for the general CA.The discussion was mainly based on the structure theorem of the finite abelian groups and the matrix associated with the global rule of the additive CA over the finite abelian p-groups.

Finite Abelian Group of | A(G) |= 27pq

we have discussed structures of Abelian group G by order |A(G) |of automoorphism group and have obtained all types of finite Abelian grooup G when the order of A(G) equals 27pq(p,q are odd primmes).

A Characterization of Metahomomorphisms on Abelian Groups

In this paper, a characterization of metahomomorphisms on Abelian groups is given.

THE SYSTEMS WITH ALMOST BANACH-MEAN EQUICONTINUITY FOR ABELIAN GROUP ACTIONS

In this paper,we present the concept of Banach-mean equicontinuity and prove that the Banach-,Weyl-and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent.Furthermore,we obtain that the topological entropy of a transitive,almost Banach-mean equicontinuous dynamical system of Abelian group action is zero.As an application of our main result,we show that the topological entropy of the Banach-mean equicontinuous system under the action of an Abelian groups is zero.

CALCULATION OF THE GIBBS DERIVATIVES ON FINITE ABELIAN GROUPS THROUGH THE DECISION DIAGRAMS

Paper considers the calculation of the values of Gibbs derivatives on finite Abelian groups. The calculation procedure is based upon the decision diagram representation of functions defined on finite Abelian groups. Approach permits processing of large functions.

Canonical Differential Calculi for Finitely Generated Abelian Groups and Their Fermionic Representations

Canonical different ial calcnli arc defined for finitely generated Abelian groups with involutions existing consistently.Two such the canonical calculi arc presented.Fermionic representations for canonical calcnli are dcfined based on quantized calculi.Fermionic representations for aforemcntioned two canonical calculi are searched out.

Non-Full Rank Factorization of Finite Abelian Groups

Tilings of p-groups are closely associated with error-correcting codes. In [1], M. Dinitz, attempting to generalize full-rank tilings of ?Zn2??to arbitrary finite abelian groups, was able to show that if p ≥5, then?Znp? admits full-rank tiling and left the case p=3, as an open question. The result proved in this paper the settles of the question for the case p=3.

An Application of Cyclotomic Polynomial to Factorization of Abelian Groups

If a finite abelian group G is a direct product of its subsets such that G = A1···Ai···An, G is said to have the Hajos-n-proprty if it follows that one of these subsets, say Ai is periodic, meaning that there exists a nonidentity element g in G such that gAi = Ai . Using some properties of cyclotomic polynomials, we will show that the cyclic groups of orders pα and groups of type (p2,q2) and (pα,pβ) where p and q are distinct primes and α, β integers ≥ 1 have this property.

Finite Abelian Groups of K3 Surfaces with Smooth Quotient

The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth.Finite groups where the quotient space are Enriques surfaces are known.In this paper,by analyzing effective divisors on smooth rational surfaces,the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth.In particular,he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian cover from a K3 surface to a Hirzebrunch surface such that the branch divisor is that effective divisor.Furthermore,he will decide the Galois group and give the way to construct that Abelian cover from an effective divisor on a Hirzebruch surface.Subsequently,he studies the same theme for Enriques surfaces.

On the Average Number of Unitary Factors of Finite Abelian Groups (Ⅱ)

Let t（G） be the number of unitary factors of finite abelian group G. In this paper we prove T（x）=∑(G≤（）t（G） =main terms+O(x(（1+2k）/（3+4k）for any exponent pair （k, 1/2+2K）. which improves on the exponent 9/25 obtained by Xiaodong Cao and the author.

On Separable Schur Rings over Abelian Groups

A finite group is said to be weakly separable if every algebraic isomorphism between two 5-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.

Splitting Extensions of Abelian by Hyper-(cyclic or finite) Groups (Ⅱ)

Let F be a locally defined formation consisting of locally solvable groups, G a hyper-（ cyclic or finite） locally solvable group and A a noetherian ZG-module with all irreducible ZG-factors being finite. The following conclusion is obtained： if G∈F, f（ ∞ ） include f（p）, f（p） ≠φ for each p∈π, and A has no nonzero F central ZG- images, then any extension E of A by G splits conjugately over A, and A has no nonzero F central ZG-factors.

On the Number of Finite Non-isomorphic Abelian Groups in Short Intervals

Let the arithmetic function a(n) denote the number of non-isomorphic Abeliangroups of order n;k, positive integer, and x≥0. We setA_k(x)= sum from n≤x a(n)=k to (1)andA_k(x;h) =A_k(x+h)-A_k(x). A. Ivice first investigated the distribution of the values of finite non-isomorphicAbelian groups in short intervals. E. Kratzel reduced the problem to estimate theerror term △(1, 2, 3;x) in the three-dimensional multiplicative problem, and furtherimproved Ivice’s result.

Number of Finite Non-isomorphic Abelian Groups in Short Intervals

Let a（n） denote the number of non-isomorphic Abelian groups of order n. For afixed integer k≥1, letAk（x, h）:=sum from n=x【n≤x+h,a（n）=k to （1）If h≥x581/1744logx=x0.33314…logx as x→∞,it was proved by A,Ivic thatAk（x, h）=（dk+o（1））h, （1）wheredk=sum from n=1 to ∞ (1/2πn integral from n=-π to π(eikt gt（n）dt≥0)),gt（n）=sum from n=d/n to (μ（n/d）eita（d）).In Ref. [2], A. Ivic and P. Shiu improved the result. They showed that if h≥x877/2653（logx）c=x0.3305…（logx）c,then Eq.（1）is true, where C is a computable constant. Based on the estimate for △（1, 2, 2;x） in Ref.[2] and elementary discussion, thisnote proves the following theorem, which gives an improvement to the problem.

Reduced groups of abelian group rings

It is well known that K0R（?）Z（?）（?）0R, where R is a commutative ring. So the Grothendieck group of R can be given by the reduced group （?）0R. On the other hand, linear representations of groups can be seen as the finitely generated projective modules over group rings. Thus, it is very useful to study the properties of reduced groups of group rings.

Fredholm operators in Toeplitz algebras associated with discrete abelian groups

The locally finite below sets are introduced and the Fredholm operators in Toeplitz algebras associated with discrete abelian groups are studied.

非次正规的非交换的非TI-子群的个数不超过21的非可解群

为了进一步研究非TI-子群的个数对有限群可解性的影响,先考察非交换的非TI?子群,证明了如果非可解群G的非交换的非TI?子群的个数不超过21,则G同构于交错群A5。进而考察非次正规的非交换的非TI-子群的个数,证明了如果非可解群G的非次正规的非交换的非TI?子群的个数不超过21,则亦有G同构于A5。

Weak Isomorphism of Finite Group

On the basis of the quasi-isomorphism of finite groups, a new mapping, weak isomorphism, from a finite group to another finite group is defined. Let G and H be two finite groups and G be weak-isomorphic to H. Then G≌H if G satisfies one of the following conditions. 1） G is a finite Abelian group. 2） The order of G is p^3. 3 ） The order of G is p^n＋1 and G has a cyclic normal subgroup N = 〈a〉 of order p^n. 4） G is a nilpotent group and if p^││G│, then for any P ∈ Sylp （G）, P has a cyclic maximal subgroup, where p is a prime; 5） G is a maximal class group of order p4（p〉3）.

On Weakly Semi-radicable Groups

In this paper, we prove that if a torsion nilpotent group G is a weak semi-radicable group, then every Sylow p-group Gp is a central-by-finite p-group, and moreover Gp's center ζ(GP) satisfies |ζ(GP) : (ζ(GP))P| <∞, that is, ζ(GP) = D×F, where D is a divisible Abelian group, and F is a finite Abelian group.

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.