\(J\) replacement and direct roots of direct powers in direct products (Q1906648)

This article is a continuation of the author's previous works and dedicated to studying the replaceability properties of the direct product of groups. Here is the main definition. Let \(J\) be the infinite cyclic group and let \(\times^n C\) denote the direct product of \(n\) copies of the group \(C\). Consider the statements (1.1) \(J\times C\approx J\times D\), (1.2) \(\times^n C\approx\times^n D\) for some positive integer \(n\) (1.3) \(G=M\times C=N\times D\), \(M\approx N\). If (1.3) implies (1.1) we say that \(M\) is \(J\) replaceable. If for a fixed group \(C\) (1.2) always implies (1.1) we say that \(C\) is root replaceable. As the author showed in his previous works a group with the maximal condition for normal subgroups is root replaceable and \(J\) replaceable. A group \(G\) is said to obey the minimal condition on direct factors if every set of direct factors of \(G\) contains a minimal member. The following statements are examples of the main results of this article. If \(G/Z(G)\) obeys the minimal condition on direct factors and if for each finite \(t\) every abelian quotient of \(\times^t G\) is root replaceable, then \(G\) is root replaceable. Suppose that abelian quotients of \(G\) are \(J\) replaceable, \(G/G'\) is root replaceable, and \(G'\) obeys the minimal condition on direct factors. Then \(G\) is root replaceable and \(J\) replaceable.