Soluble groups which are products of nilpotent minimax groups

On Normal Subgroups of Products of Nilpotent Groups

“…More generally the first author proved in [1] that the same conclusion holds for soluble products G = AB of two nilpotent subgroups A and B with A =£ B, provided that one of them satisfies the maximum or minimum condition on subgroups. In fact a similar argument applies if one of the two factors A and B is minimax (see [5]). …”

Soluble Groups Which are Products of Groups of Finite Rank

“…We begin by quoting a theorem of Zaicev. The first assertion is a slightly extended version of [10, Theorem 1] (see also [1,Lemma 2]), and the second assertion follows from the first on considering the split extension G = MxH, which is the product of its subgroups H and {h(hS):heH}.…”

“…Condition (i) holds from step (c) above, condition (ii) since C M (D) is an F//-submodule and H acts faithfully on M. Condition (iii) holds by M(3.4)(a). By M(3.3) we have N = (S}S, where N is the <5>-module generated by N. If \H:<5>| is infinite we have r o ((S)) < r Q (H), so that Nis finite-dimensional; by (3,1) it follows that N = 0, so that DS = 0 and D ^ C H (M) = 1, and this is a contradiction. Therefore condition (iv) holds.…”

On groups with a triple factorisation

“…However, in the final corollary we are able to give a surprisingly short proof, thereby avoiding quite heavy tools such as cohomology theory and splitting theorems for infinite groups (see [10] …”