A Note on the Abelianization Functor

It is well known that the abelianization of a group G can be computed as the cokernel of the diagonal morphism (1_G, 1_G): G → G × G in the category of groups. We generalize this to arbitrary regular subtractive categories, among which are the category of groups, the category of topological groups, and the categories of other group-like structures. We also establish that an abelian category is the same as a regular subtractive category in which every monomorphism is a kernel of some morphism.     

Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groups

In this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph Γ is n‐HC‐extendable if it contains a path of length n and if every such path is contained in some Hamilton cycle of Γ. Similarly, Γ is weakly n‐HP‐extendable if it contains a path of length n and if every such path is contained in some Hamilton path of Γ. Moreover, Γ is strongly n‐HP‐extendable if it contains a path of length n and if for every such path P there is a Hamilton path of Γ starting with P. These concepts are then studied for the class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley graph on an abelian group of order at least three is 2‐HC‐extendable and a complete classification of 3‐HC‐extendable connected Cayley graphs of abelian groups is obtained. Moreover, it is proved that every connected Cayley graph on an abelian group of order at least five is weakly 4‐HP‐extendable. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory     

On Cayley digraphs on nonisomorphic 2‐groups

A necessary and sufficient condition is given for two Cayley digraphs X₁ = Cay(G₁, S₁) and X₂ = Cay(G₂, S₂) to be isomorphic, where the groups G_i are nonisomorphic abelian 2‐groups, and the digraphs X_i have a regular cyclic group of automorphisms. Our result extends that of Morris [J Graph Theory 3 (1999), 345–362] concerning p‐groups G_i, where p is an odd prime. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory     

Finite p-Groups with a Cyclic Subgroup of Index p3

We classify up to isomorphism those finite p-groups,for odd primes p,which contain a cyclic subgroup of index p3.     

A generalization of Meshulam’s theorem on subsets of finite abelian groups with no 3-term arithmetic progression

Let r ₁, …, r _s be non-zero integers satisfying r ₁ + ⋯ + r _s = 0. Let G be a finite abelian group with k _i |k _i-1(2 ≤ i ≤ n), and suppose that (r _i , k ₁) = 1(1 ≤ i ≤ s). Let denote the maximal cardinality of a set which contains no non-trivial solution of r ₁ x ₁ + ⋯ + r _s x _s = 0 with . We prove that . We also apply this result to study problems in finite projective spaces.      

From recollement of triangulated categories to recollement of abelian categories

In this paper,we prove that if a triangulated category D admits a recollement relative to triangulated categories D' and D″,then the abelian category D/T admits a recollement relative to abelian categories D'/i(T) and D″/j(T) where T is a cluster tilting subcategory of D and satisfies i i (T)  T,j j (T) T.     

有限温度阿贝尔格点规范理论相变的解析研究

应用交分累积展开方法系统地研究了（２＋１）维和（３＋１）维的Ｚ（２）、Ｚ（３）、Ｕ（１）阿贝尔格点规范理论在有限温度下的相交．所得结果同已有的严格结果及Ｓｖｅｔｉｔｓｋｙ和Ｙａｆｆ的理论预言相符合．     

New results on non-existence of generalized bent functions (II)

Several new results on non-existence of generalized bent functions are presented by using the class group of related imaginary abelian number fields.