On the residual nilpotence of free products of free groups with cyclic amalgamation
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 \(p\).
residually finite solvable \(\pi\)-groups
scientific article; zbMATH DE number 1274209
scientific article