Soluble products of minimax groups, and nearly surjective derivations (Q1108378)

The author proves that a soluble group \(G=AB\) which is the product of two minimax subgroups \(A\) and \(B\) is a minimax group. (A soluble group is a minimax group if it has a finite series whose factors are cyclic or Prüfer groups.) This solves Problem 9.54 of the Kourouvka Notebook [1986; Zbl 0625.20001]. Some special cases of this result where one of the two subgroups \(A\) or \(B\) satisfies a nilpotency requirement, were previously dealt with by D. I. Zaicev, D. Robinson and the reviewer. For the proof of the above theorem one needs to show that whenever \(\beta\) is a (nearly) surjective derivation from a soluble minimax group \(H\) to an \(H\)-module \(M\), then, regarded as a group, \(M\) is minimax. (Such a derivation is called nearly surjective if the set \(H\beta\) has a finite complement in \(M\).) After some reductions one has to prove that there is no simple module \(M\) for a soluble minimax group \(H\) with the property that there is a (nearly) surjective derivation from \(H\) to \(M\).