Generalized free products of \(\pi_ c\) groups
A group \(G\) is said to be \(\pi_ c\) if and only if for every pair of elements \(g_ 1\) and \(g_ 2\) of \(G\) either \(g_ 1= g_ 2^ k\) for some integer \(k\) or there exists a normal subgroup \(N\) of finite index in \(G\) such that \(g_ 1\not\equiv g_ 2^ m\bmod N\) for all integers \(m\). A group \(G\) is said to be \((x)\)-Pot for a nontrivial element \(x\) of \(G\) if and only if for every positive integer \(n\) (dividing the order of \(x\) if the order is finite) there exists a normal subgroup \(N\) of finite index \(G\) such that \(xN\) has order exactly \(n\) in \(G/N\). Let \(A\) and \(B\) be \(\pi_ c\) groups and let \(a\in A\) and \(b\in B\) where \(a\) and \(b\) have the same order. The authors prove that if \(A\) is \((a)\)- Pot and \(B\) is \((b)\)-Pot, then the generalized free product \(G\) of \(A\) and \(B\) amalgamating the subgroups \((a)\) and \((b)\) with \(a=b\), is \(\pi_ c\).
scientific article; zbMATH DE number 219354