[Paper Review] On the cyclic subgroup separability of free products of two groups with amalgamated subgroup

E. V. Sokolov|ArXiv.org|Aug 21, 2007

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper establishes sufficient conditions for the $p$-separability of $p'$-isolated cyclic subgroups in free products of groups with cyclic amalgamated subgroups. It proves that if the free factors are free or finitely generated residually $p$-finite nilpotent groups, then residual $p$-finiteness of the free product implies $p$-separability of all $p'$-isolated cyclic subgroups, extending known results on subgroup separability in amalgamated products.

ABSTRACT

Let $G$ be a free product of two groups with amalgamated subgroup, $π$ be either the set of all prime numbers or the one-element set \{$p$\} for some prime number $p$. Denote by $Σ$ the family of all cyclic subgroups of group $G$, which are separable in the class of all finite $π$-groups. Obviously, cyclic subgroups of the free factors, which aren't separable in these factors by the family of all normal subgroups of finite $π$-index of group $G$, the subgroups conjugated with them and all subgroups, which aren't $π^{\prime}$-isolated, don't belong to $Σ$. Some sufficient conditions are obtained for $Σ$ to coincide with the family of all other $π^{\prime}$-isolated cyclic subgroups of group $G$. It is proved, in particular, that the residual $p$-finiteness of a free product with cyclic amalgamation implies the $p$-separability of all $p^{\prime}$-isolated cyclic subgroups if the free factors are free or finitely generated residually $p$-finite nilpotent groups.

Motivation & Objective

• To determine conditions under which all $p'$-isolated cyclic subgroups of a free product with cyclic amalgamated subgroup are $p$-separable.
• To extend known results on cyclic subgroup separability to the case of free products with amalgamated subgroups.
• To characterize the family of cyclic subgroups that are separable in the class of finite $p$-groups within such amalgamated products.
• To investigate the relationship between $p$-finiteness of the whole group and $p$-separability of its $p'$-isolated cyclic subgroups.
• To generalize the notion of $H$-filtrations and $K$-filtrations to establish sufficient conditions for separability in the amalgamated product setting.

Proposed method

• The paper uses the concept of $H$-filtrations and $K$-filtrations in the free factors $A$ and $B$ to analyze separability in the amalgamated product $G = A*_H B$.
• It defines families $\Omega_A$ and $\Omega_B$ of normal subgroups of finite index that are $(H,K,\varphi)$-compatible, and uses them to characterize separability of cyclic subgroups.
• The proof relies on analyzing elements that are not mapped to the identity under any homomorphism to finite $p$-groups, particularly focusing on commutators and centralizers.
• It applies the property that if a subgroup is not separable in the class of finite $p$-groups, then there exists an element outside the subgroup that maps into it under all such homomorphisms.
• The argument uses the structure of cyclic subgroups in $G$, especially those conjugate to subgroups in $A$ or $B$, and examines their behavior under homomorphisms to finite $p$-groups.
• It leverages the fact that $p'$-isolated subgroups are those whose order is coprime to $p$, and uses this to rule out nontrivial commutators that would contradict residual $p$-finiteness.

Experimental results

Research questions

• RQ1 Under what conditions is every $p'$-isolated cyclic subgroup of a free product with cyclic amalgamation $p$-separable?
• RQ2 How does the residual $p$-finiteness of the amalgamated product relate to the $p$-separability of its $p'$-isolated cyclic subgroups?
• RQ3 What role do $H$-filtrations and $K$-filtrations in the free factors play in ensuring separability of cyclic subgroups in the amalgamated product?
• RQ4 Can the $p$-separability of $p'$-isolated cyclic subgroups be guaranteed when the free factors are free or finitely generated residually $p$-finite nilpotent groups?
• RQ5 What structural constraints on the amalgamated subgroup and the free factors prevent nontrivial commutators from being trivialized under all $p$-group homomorphisms?

Key findings

• If the free factors $A$ and $B$ are free or finitely generated residually $p$-finite nilpotent groups, then residual $p$-finiteness of $G$ implies $p$-separability of all $p'$-isolated cyclic subgroups.
• The family of $p$-separable $p'$-isolated cyclic subgroups coincides with the family of all such subgroups not conjugate to any subgroup in $\Lambda_A^p \cup \Lambda_B^p$, where $\Lambda_A^p$ and $\Lambda_B^p$ are the families of cyclic subgroups not separable by the subgroups of $\Omega_A^p$ and $\Omega_B^p$.
• The condition that $\Omega_A$ is an $H$-filtration and $\Omega_B$ is a $K$-filtration is sufficient for the $p$-separability of all $p'$-isolated cyclic subgroups not conjugate to elements in $\Lambda_A^p \cup \Lambda_B^p$, generalizing earlier results on residual finiteness.
• If a cyclic subgroup $\langle u \rangle$ of $G$ is $p'$-isolated but not $p$-separable, then any element $v$ such that $v\psi \in \langle u\psi \rangle$ for all $\psi$ in the family of homomorphisms to finite $p$-groups must commute with $u$, leading to a contradiction unless $u$ is conjugate to a subgroup in $A$ or $B$.
• The paper shows that if $b^{-1}Kb \cap K \neq 1$ for $b \in B \setminus K$, then a nontrivial commutator $g = [b^{-1}f_n b, f_n]$ can be constructed that maps to the identity under all $p$-group homomorphisms, contradicting residual $p$-finiteness.
• The proof concludes that $b^{-1}Kb \cap K = 1$ for all $b \in B \setminus K$, which is a necessary condition for the $p$-separability of $p'$-isolated cyclic subgroups in $G$.