Root-class residuality of fundamental group of a finite graph of group

D. V. Goltsov

Ivanovo State University

Abstract: Let $\mathcal{K}$ be an abstract class of groups. Suppose $\mathcal{K}$ contains at least a non trivial group. Then $\mathcal{K}$ is called a root-class if the following conditions are satisfied:
1. If $A \in \mathcal{K}$ and $B \leq A$, then $B \in \mathcal{K}$.
2. If $A \in \mathcal{K}$ and $B \in \mathcal{K}$, then $A\times B \in \mathcal{K}$.
3. If $1\leq C \leq B \leq A$ is a subnormal sequence and $A/B, B/C \in \mathcal{K}$, then there exists a normal subgroup $D$ in group $A$ such that $D \leq C$ and $A/D \in \mathcal{K}$.
Group $G$ is root-class residual (or $\mathcal{K}$-residual), for a root-class $\mathcal{K}$ if, for every $1 \not = g \in G$, exists a homomorphism $\varphi $ of group $G$ onto a group of root-class $\mathcal{K}$ such that $g\varphi \not = 1$. Equivalently, group $G$ is $\mathcal{K}$-residual if, for every $1 \not = g \in G$, there exists a normal subgroup $N$ of $G$ such that $G/N \in \mathcal{K}$ and $g \not \in N$. The most investigated residual properties of groups are finite groups residuality (residual finiteness), $p$-finite groups residuality and soluble groups residuality. All there three classes of groups are root-classes. Therefore results about root-class residuality have safficiently enough general character.
Let $\mathcal{K}$ be a root-class of finite groups. And let $G$ be a fundamental group of a finite graph of groups with finite edges groups. The necessary and sufficient condition of virtual $\mathcal{K}$-residuality for the group $G$ is obtained.
Bibliography: 16 titles.

Keywords: root-class of finite groups, fundamental group of a finite graph of groups, virtual $\mathcal{K}$-residuality.

UDC: 512.543

Received: 03.06.2016
Accepted: 13.09.2016

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