He Guo LIU, Jing ZHAO

Constructing examples of groups is an important aspect in the theory of groups. We will study the residual finiteness of two concrete matrix groups. Let $p$ be a prime, let $C=\langle c\rangle$ be an infinite cyclic group, let $R=\mathbb{Z}C$ be the integral group ring over $C$, and let $U(n,R)$ be the upper unitriangular group over $R$ of order $n$, where $n\geq 2$, which is a nilpotent group of infinite rank of class $n-1$. Firstly, we prove that $U(n,R)$ is a residually finite $p$-group. Secondly, let $ G=\langle\alpha\rangle\ltimes U(3,R)$, where $\alpha={\rm diag}(c,1,c)$ is a diagonal matrix of order 3. We will study the structure of $G$ and prove that $G$ is a residually finite $p$-group, $G$ is a 3-generated soluble group of derived length 3. Moreover, we will construct two quotient groups of $G$, neither of which is residually finite. These two quotient groups seem to be more elementary and concrete than the classical examples discovered by Hall.