A new alternating group explicit-implicit algorithm with high accuracy for dispersive equation

In this paper, a new alternating group explicit-implicit （nAGEI） scheme for dispersive equations with a periodic boundary condition is derived. This new unconditionally stable scheme has a fourth-order truncation error in space and a convergence ratio faster than some known alternating methods such as ASEI and AGE. Comparison in accuracy with the AGEI and AGE methods is presented in the numerical experiment.

Important Properties of Alternating Group A_6

By using electric computer machine, via computation method, we obtained some important properties of alternating group A \{6\} as below: 1) A \{6 \} has 501 subgroups in total, and for each subgroup we give its generators; 2) the index of A \{6\}'s subgroup can only be 1,2,3,4,5,6,8,9,10,12,18,24,36,60,360; 3) all subgroups of A \{6\} are separated into 22 conjugate classes, and the subgroups contained in each class are listed.

A CLASS OF ALTERNATING GROUP METHOD OF BURGERS' EQUATION

Some new Saul'yev type asymmetric difference schemes for Burgers' equation is given, by the use of the schemes, a kind of alternating group four points method for solving nonlinear Burgers' equation is constructed here. The basic idea of the method is that the grid points on the same time level is divided into a number of groups, the difference equations of each group can be solved independently, hence the method with intrinsic parallelism can be used directly on parallel computer. The method is unconditionally stable by analysis of linearization procedure. The numerical experiments show that the method has good stability and accuracy.

Automorphism Groups of a Family of Cayley Graphs on Alternating Groups

Let W = {ω1,ω2,…,ωn-1} be a minimal generating transposition set of Sn. In this paper it was shown that V = （0, p）W = {（0, p）ω1, （0, p）ω2…, （0,p）ωn-1} is a minimal generating set of alternating group An＋1, where p ∈{1, 2, …, n}. And then we investigated the automorphism groups of Cayley graphs on alternating groups with these generating sets which contain alternating group graph AGn.

QUASI-PERMUTATION REPRESENTATIONS OF ALTERNATING AND SYMMETRIC GROUPS

The quantities c（G）, q（G） and p（G） for finite groups were defined by H. Behravesh. In this article, these quantities for the alternating group An and the symmetric group Sn are calculated. It is shown that c（G） = q（G） = p（G） = n,when G = An or Sn.

Single Alternating Group Explicit (SAGE) Method for Electrochemical Finite Difference Digital Simulation

The four different schemes of Group Explicit Method (GEM): GER, GEL, SAGE andDAGE have been claimed to be unstable when employed for electrochemical digital simulations withlarge model diffusion coefficient D_M. However, in this investigation, in spite of the conditionalstability of GER and GEL, the SAGE scheme, which is a combination of GEL and GER, was found to beunconditionally stable when used for simulations of electrochemical reaction-diffusions and had aperformance comparable with or even better than the Fast Quasi Explicit Finite Difference Method(FQEFD) in some aspects. Corresponding differential equations of SAGE scheme for digital simulationsof various electrochemical mechanisms with both uniform and exponentially expanded space units wereestablished. The effectiveness of the SAGE method was further demonstrated by the simulations of anEC and a catalytic mechanism with very large homogeneous rate constants.

AN EXPONENTIAL-TYPE WEIGHTED IMPLICIT SCHEME AND ALTERNATING GROUP EXPLICIT ITERATIVE METHOD FOR BURGERS EQUATION

A new exponential-type weighted implicit difference schcme for solving the Burgers equation was developed and iterated by using the Alternating Group Explicit （AGE） method which has the obvious property of parallelism. The stability of the method was analyzed with linearization procedure. A model problem was numerically solved with the proposed scheme. The results show that the method is superb or to some existing schemes.

On Huppert＇s Conjecture for Alternating Groups of Low Degrees

Bertram Huppert conjectured in the late 1990s that the nonabelian simple groups are determined up to an abelian direct factor by the set of their character degrees. Although the conjecture has been established for various simple groups of Lie type and simple sporadic groups, it is expected to be difficult for alternating groups. In [5], Hup- pert verified the conjecture for the simple alternating groups AN of degree up to 11. In this paper, we continue his work and verify the conjecture for the alternating groups of degrees 12 and 13.

Finite Presentation of Alternating Groups

Graham Higman posed the question: How small can the integers p and q be made, while maintaining the property that all but finitly many alternating and symmetric groups are factor groups of △(2, p, q)=(x,y: x^2=y^P=(xy)~q=1)? He proved that for a sufficiently large n, the alternating group is a homomorphic image of the triangle group △(2,p, q) where p=3 and q=7. Later, his result was generalized by proving the result for p=3 and q≥7. Choosing p=4 and q≥17 in this paper we have answered the "Hiqman Question".

Group-Theoretic Remarks on Goldbach’s Conjecture

The famous strongly binary Goldbach’s conjecture asserts that every even number 2n ≥ 8 can always be expressible as a sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we apply the element order prime graphs of alternating groups of degrees 2n and 2n −1 to characterize this conjecture, and present its six group-theoretic versions;and further prove that this conjecture is true for p +1 and p −1 whenever p ≥ 11 is a prime number.

OD-Characterization of Alternating and Symmetric Groups of Degree p+5

In this paper,it is proved that all the alternating groups A_(p+5) are ODcharacterizable and the symmetric groups S_(p+5) are 3-fold OD-characterizable,where p + 4 is a composite number and p + 6 is a prime and 5≠p∈π(1000!).

Connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups

A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two sufficient and necessary conditions for such graphs to be 1- or 2-arcregular are given and based on the conditions, several infinite families of 1- or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.

Integral Cayley Graphs over Finite Groups

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group(s)is integral.In particular,a Cayley graph of a 2-group generated by a normal set of involutions is integral.We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral.We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form(k i j)with fixed k,a s{-n+1,1-n+1,2^2-n+1,...,(n-1)2-n+1}.

Flag-transitive 2-（v, k, λ） symmetric designs with （k, λ） =1 and alternating socle

Consider the flag-transitive 2-（v, k, λ） symmetric designs with （k, λ）= 1. We prove that if （） is a nontrivial 2-（v, k, λ）symmetric design with （k,λ） = 1 and G ≤ Aut（） is flag-transitive with Sot（G） = An for n ≥5, then is the projective space PG2（3, 2） and G = A7.