Some corollaries of Frobenius’ normal $p$-complement theorem

Abstract:

For a prime divisor $q$ of the order of a finite group $G$, we present the set of $q$-subgroups generating $\text {O}^{q,q’}(G)$. 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.