Some corollaries of Frobenius’ normal 𝑝-complement theorem

Abstract

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