Narrow Results

For a positive integer $n\ge 4$, let $R_n$ be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank n. We show that the subgroup of $\mathrm{\text{Aut}}(R_n)$ generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of $R_n$ is dense in $\mathrm{\text{Aut}}(R_n)$ with respect to the formal power series topology.

The orthogonality graph$O(R)$ of a ring $R$ is the undirected graph with vertex set consisting of all nonzero two-sided zero divisors of $R$, in which for two vertices $x$ and $y$ (needless distinct), $x\sim y$ is an edge if and only if $xy=yx=0$. Let $n\ge 2$, ${\mathrm{\text{Mat}}}_n(F)$ be the set of all $n\times n$ matrices over a finite field $F$, and $R_n(F)$ the subset of ${\mathrm{\text{Mat}}}_n(F)$ consisting of all rank one upper triangular matrices. In this paper, we describe the full automorphism group, and using the technique of generalized equivalent canonical form of matrices, we compute the fixing number of $O(R_n(F))$, the induced subgraph of $O({\mathrm{\text{Mat}}}_n(F))$ with vertex set $R_n(F)$.

A subset $D$ of a group $G$ is a determining set of $G$ if every automorphism of $G$ is uniquely determined by its action on $D$, and the determining number of $G$, $\alpha (G)$, is the cardinality of a smallest determining set. A group $G$ is called a DEG-group if $\alpha (G)$ equals $\gamma (G)$, the generating number of $G$. Our main results 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; for a given finite group $H$, there is a DEG-group $G$ such that $H$ is isomorphic to a normal subgroup of $G$ and there is an injective mapping from the set of all finite groups to the set of finite DEG-groups; for any integer $k\ge 2$, there exists a group $G$ such that $\alpha (G)=2$ and $\gamma (G)\ge k$.

In this paper, we consider the automorphisms of the strong semilattice of groups and relate them to the isomorphisms and automorphisms of underlying groups. We also provide a construction for non-trivial automorphisms of semilattices.

In this paper, we give an explicit expression for the number of pairwise non-isomorphic connected regular graphs as well as odd graphs on n vertices, where $n\ge 2$ is a positive integer.