This article is cited in 15 papers

On the root-class residuality of certain free products of groups with normal amalgamated subgroups

E. V. Sokolov, E. A. Tumanova

Ivanovo State University, 39 Ermak str., Ivanovo, 153025 Russia

Abstract: Let $\mathcal{K}$ be a root class of groups closed under taking quotient groups, $G$ be a free product of groups $A$ and $B$ with amalgamated subgroups $H$ and $K$. Let also $H$ be normal in $A$, $K$ be normal in $B$, and $\operatorname{Aut}_{G}(H)$ denote the set of automorphisms of $H$ induced by all inner automorphisms of $G$. We prove a criterion for $G$ to be residually a $\mathcal{K}$-group provided $\operatorname{Aut}_{G}(H)$ is an abelian group or it satisfies some other conditions. We apply this result in the cases when $A$ and $B$ are bounded nilpotent groups or $A/H, B/K \in \mathcal{K}$.

Keywords: generalized free product, residual finiteness, residual $p$-finiteness, residual solvability, root-class residuality.

UDC: 512.543

Received: 11.03.2019
Revised: 25.04.2019
Accepted: 19.06.2019

DOI: 10.26907/0021-3446-2020-3-48-63

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2020, 64:3, 43–56

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