What is your motivation to choose subgroups of derived subgroup?

@StefanKohl It is known that if $6\not\in {\rm P}_3(G)$ then $G$ is metabelian and if $|G'|\leq 5$ then $6\not\in {\rm P}_3(G)$.

@StefanKohl In the definition of $P_3(G)$ there is something wrong?! Please check it; you may do a swap ${\rm P}_3(G) := \left\{ \ | \{abc, acb, bac, bca, cab, cba\} | \ |\ a,b,c \in G \right\}$.

See the following post mathoverflow.net/questions/85540/…

If $\mathcal{P}$ is a group-theoretical property then a group is called weakly $\mathcal{P}$ if every 2-generator subgroup is a $\mathcal{P}$-group. See page 38 of [Finiteness Conditions and Generalized Soluble Groups, Part 2 By Derek J.S. Robinson, Springer ,1972] to find "weakly nilpotent". Groups which are infinite weakly finite share some properties as locally finite groups: e.g. they all have infinite abelian subgroups.

@Maryam The condition ``$x$ normalize $A$" is superfluous, since in nilpotent groups maximal abelian subgroups are normal.

@scalar Please let me know what are the "very little information" that you have have found so far.

What is $M(R)$? What is $M(R')$? Are they obvious misprint? I.e they must be