Some corollaries of Frobenius' normal p-complement theorem

For a prime divisor q of the order of a finite group G, we present the set of q-subgroups generating O-q,O-q' (G). In particular, we present the set of primary subgroups of G generating the last member of the lower central series of G. The proof is based on the Frobenius Normal p-Complement Theorem and basic properties of minimal nonnilpotent groups. Let G be a group and Theta a group-theoretic property inherited by subgroups and epimorphic images such that all minimal non-Theta-subgroups (= Theta(1)-subgroups) of G are not nilpotent. Then (see the lemma), if K is generated by all Theta(1)-subgroups of G it follows that G/K is a Theta-group.