On finite soluble groups verifying an extremal condition on subgroups (Q1210113)

The author introduces the concept of an \(F\)-extremal. A finite soluble group \(G\) is called \(F\)-extremal when for every non-nilpotent subgroup \(H\) then \(C_ G(H')=F(H)\) where \(F(H)\) is the Fitting subgroup of \(H\). The main result is the following theorem: \(G\) is \(F\)-extremal if and only if one of the following two conditions holds: (i) \(G\) is nilpotent; (ii) \(G\) is an \(F\)-extremal \(E\)-extension, i.e. a metabelian, non-nilpotent, central extension by a group all of whose Sylow subgroups are abelian.