Groups with weak minimality and maximality conditions for subgroups which are not normal (Q1812794)

If \(\mathfrak X\) is a group-theoretical property, a group \(G\) is said to satisfy the weak minimal condition for \(\mathfrak X\)-subgroups (respectively: the weak maximal condition for \(\mathfrak X\)-subgroups) if \(G\) does not have any infinite strictly descending chain of \(\mathfrak X\)-subgroups \(H_ 1>H_ 2>\dots\) in which all indices \(| H_ n: H_{n+1}|\) are infinite (respectively: any infinite strictly ascending chain of \(\mathfrak X\)-subgroups \(H_ 1<H_ 2<\dots\) in which all indices \(| H_{n+1}: H_ n|\) are infinite). Here the authors prove that a (locally- soluble)-by-finite group either satisfying the weak minimal condition for non-normal subgroups or the maximal condition for non-normal subgroups is soluble-by-finite, and either is a Dedekind group or a minimax group.