On a Class of Residually Finite Groups

Taeri, B., (2003) On a Class of Residually Finite Groups. Bulletin of the Malaysian Mathematical Sciences Society, 26 (2). pp. 209-219. ISSN 0126-6705

Full text not available from this repository.

Affiliations

Isfahan University of Technology, Dept. of Mathematics

Abstract

Let $n$ , $k$ be positive integers and ${t}_0,{t}_1,...,{t}_k$ be non-zero integers. We denote by $\overline{W}_k(n)$ the class of groups ${G}$ in which, for every subset ${X}$ of ${G}$ of cardinality $n + 1$ , there exist a subset ${X}_0\subseteq{X}$ , with $2 \leq |{X}_0| \leq {n} + 1$ ,, and a function ${f} : {0,1,2,...,k} \rightarrow {X}_0$ , with $f(0) \neq f(1)$ such that [ $x_0^{t_0}$ , $x_1^{t_1}$ ,..., $x_k^{t_k}$ ] $= 1$ where $x_i := f(i)$ , $i = 0,1,...,k$ . The class $\overline{W}^\ast_k(n)$ is defined exactly as $\overline{W}_k(n)$ , with additional conditions " $x_j \in H$ whenever $x_j ^{t_j} \in H$ , where $\langle x_j^{t_j}\rangle \neq H \leq G$ “.

Let $G$ be a finitely generated residually finite group. Here we prove that

(1) If $G \in \overline{W}_k(n)$ , then $G$ has a normal nilpotent subgroup $N$ with finite index such that the nilpotency class of $N / N_t$ , is bounded by a function of $k$ , where $N_t$ , is the torsion subgroup of $N$ .

(2) If $G \in \overline{W}_k^\ast(n)$ be $d$ generated, then $G$ has a normal nilpotent subgroup $N$ whose index and the nilpotency class are bounded by a function of $k,n,t_0,t_1,...,t_k$ .

Item Type:

Journal

Subjects:

Q Science, Computer Science

ID Code:

1195