Nilpotent complements and Carter subgroups in stable \({\mathfrak R}\) (Q1322452)

The author continues the approach of his paper [Notre Dame J. Formal Logic 33, 159-174 (1992; Zbl 0805.03022)]. An \({\mathcal R}\)-theory is a stable theory such that for all definable transitive group actions of some definable group \(G\) on some definable set \(X\), if \(x\in X\) is generic for that action and \(x\) is algebraic over some element \(y\in X\), then \(y\) is generic as well. An \({\mathcal R}\)-group is a group whose theory is an \({\mathcal R}\)-theory. The following theorems are proved about the Frattini-free component \(G^ \Phi\) of a soluble stable \({\mathcal R}\)-group: 1) If it has a normal subgroup \(N\) with nilpotent quotient, then there is a nilpotent subgroup \(H\) of \(G^ \Phi\) with \(G^ \Phi= NH\). 2) It has Carter subgroups (i.e. nilpotent self-normalising subgroups); if \(G\) is small, they are all conjugate. 3) Nilpotency modulo a suitable Frattini- subgroup implies nilpotency (of \(G^ \Phi\)).