On the residual nilpotence of generalized free products of groups (Q6588194)

Let \(G=\langle A, B \ ; \ H \rangle\) be the generalized free product of groups \(A\) and \(B\) with an amalgamated subgroup \(H\), and let \(\mathfrak{P}\) be a set of primes. This paper presents a general approach that allows the use of results on the residual \(p\)-finiteness of \(G\) to study the residual \(\mathcal{C}\)-ness of \(G\), where \(\mathcal{C}\) is the class of finite nilpotent or finite metanilpotent groups. It demonstrates that this approach can be applied under each of the restrictions on \(H\) that allow the study of residual \(p\)-finiteness.\N\NThe main results deal with \(G\) and a set of primes \(\mathfrak{P}\) satisfying the following conditions:\N\N(i) \(A \neq H \neq B\);\N\N(ii) \(\mathfrak{P}\) is a non-empty set;\N\N(iii) \(A\) and \(B\) are residually \(\mathcal{FN}_{\mathfrak{P}}\)-groups;\N\N(iv) there exist homomorphisms of the groups \(A\) and \(B\) that map them onto \(\mathcal{BN}_{\mathfrak{P}}\)-groups and act injectively on \(H\).\N\NFinite nilpotent \(\mathfrak{P}\)-groups and \(\mathfrak{P}\)-bounded nilpotent groups are denoted by \(\mathcal{FN}_{\mathfrak{P}}\)-groups and \(\mathcal{BN}_{\mathfrak{P}}\)-groups, respectively.\N\NTheorems 1--7 address the cases where \(G\) and a set of primes \(\mathfrak{P}\) satisfies (i)--(iv), and simultaneously, \(H\) is locally cyclic, lies in the center of \(A\) or \(B\), is a retract of \(A\) or \(B\), or is normal, periodic, or nilpotent, demonstrating that this approach can be applied under each of these conditions on \(H\) that allow the study of residual \(p\)-finiteness.\N\NEspecially, Theorem 7 addresses the main tool of this paper, which allows one to use results on the residual \(p\)-finiteness of the group \(G\) to study the residual nilpotence of \(G\).\N\NAdditionally, the two Examples 1 and 2 provided in this paper specifically apply the conditions of Theorems, explaining the necessity of these conditions and implying that they can be practically used to understand the properties discussed in the paper for the given groups with specific presentations. Moreover, Examples 1 and 2 explain the necessary conditions regarding the properties of \(H\) and \(G\) in each theorem and analyze how these conditions are connected to one another.