arXiv subjects
Publications and source records attributed to Dengyin Wang.
In this paper, we introduce the weakly nilpotent hypergroups with giving some new properties, and then establish several structural characterizations of these hypergroups. Some results obtained in this paper answer the two questions raised by the first author and W. Guo.
Let $G$ be a group. A subset $D$ of $G$ is a determining set of $G$, if every automorphism of $G$ is uniquely determined by its action on $D$. The determining number of $G$, denoted by $α(G)$, is the cardinality of a smallest determining set. A generating set of $G$ is a subset such that every element of $G$ can be expressed as the combination, under the group operation, of finitely many elements of the subset and their inverses. The cardinality of a smallest generating set of $G$, denoted by $γ(G)$, is called the generating number of $G$. A group $G$ is called a DEG-group if $α(G)=γ(G)$. The main results of this article are as follows. Finite groups with determining number $0$ or $1$ are classified; Finite simple groups and finite nilpotent groups are proved to be DEG-groups; A finite group is a normal subgroup of a DEG-group and there is an injective mapping from the set all finite groups to the set of finite DEG-groups; Nilpotent groups of order $n$ which have the maximum determining number are classified; For any integer $k\geq 2$, there exists a group $G$ such that $α(G)=2$ and $γ(G)\geq k$.