On the residual nilpotence of free products of free groups with cyclic amalgamation
type: Item
instance of: scholarly article
title: On the residual nilpotence of free products of free groups with cyclic amalgamation
zbMATH Open document ID: 0922.20030
DOI: 10.1007/BF02307189
author: D. N. Azarov
published in: Mathematical Notes
publication date: 1999-10-19T00:00:00Z
review text: Let \(G=(A*B;\;h=k)\) be the free product of free groups with cyclic amalgamation, \(h\neq 1\), \(k\neq 1\). Let \(m\) and \(n\) be the largest positive integers such that the equations \(x^m=h\) and \(y^n=k\) are solvable in the groups \(A\) and \(B\), respectively. The author proves that if one of the numbers \(m\) and \(n\) is 1, then for any prime \(p\) the group \(G\) is a residually finite \(p\)-group. He shows that if \(m>1\) and \(n>1\) then the group \(G\) is a residually finite \(p\)-group if and only if the numbers \(m\) and \(n\) are powers of the prime \(p\). Also he finds a criterion for a group \(G\) to be a residually finite solvable \(\pi\)-group. The author establishes that a group \(G\) is residually nilpotent if and only if it is a residually finite \(p\)-group for an appropriate prime \(p\).
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E06
zbMATH DE Number: 1274209
zbMATH Keywords: free products of free groups
zbMATH Keywords: residually finite \(p\)-groups
zbMATH Keywords: residually finite solvable \(\pi\)-groups
zbMATH Keywords: residually nilpotent groups
reviewed by: Alexander Ivanovich Budkin
MaRDI profile type: Publication
cites work: On the Residual Finiteness of Generalised Free Products of Nilpotent Groups
cites work: Q3866296
cites work: Q3688535
cites work: One-relator groups that are residually of prime power order
cites work: Conjugacy Separability of Certain Free Products with Amalgamation
cites work: Residual Properties of Infinite Soluble Groups
cites work: Amalgams of \(p\)-groups.
cites work: On the residual nilpotence of certain one‐relator groups
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description: scientific article; zbMATH DE number 1274209