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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.
Received: 03.06.2016 Accepted: 13.09.2016
Citation:
D. V. Goltsov, “Root-class residuality of fundamental group of a finite graph of group”, Chebyshevskii Sb., 17:3 (2016), 64–71
Linking options:
https://www.mathnet.ru/eng/cheb498 https://www.mathnet.ru/eng/cheb/v17/i3/p64
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