Groups with \(\pi\)-maximal and \(\pi\)-layer maximal conditions (Q5930799)

Let \(\pi\) be a set of prime numbers. A group \(G\) is said to satisfy the \(\pi\)-maximal condition if it has no infinite ascending series \(G_1<G_2<\cdots<G_i<\cdots\) of subgroups such that for each \(i\) the difference \(G_{i+1}\setminus G_i\) contains a \(\pi\)-element. Moreover, \(G\) is said to satisfy the \(\pi\)-layer maximal condition if for each positive integer \(n\) it has no infinite ascending series \(G_1<G_2<\cdots<G_i<\cdots\) of subgroups such that for each \(i\) the difference \(G_{i+1}\setminus G_i\) contains a \(\pi\)-element with order at most \(n\). Some descriptions of groups satisfying the \(\pi\)-maximal condition or the \(\pi\)-layer maximal condition are given.