Let $G$ be a $p$-group for an odd prime $p$. The Frattini subgorup $\Phi(G)$ is defined as the intersection of all maximal subgroups or, equivalently, as the subgroup of non-generating elements, i.e. the elements $x \in G$ such that for every $R \subseteq G$ with $\langle x,R\rangle=G$ we must have that $G=\langle R\rangle$. The Agemo subgroups $\mho_k(G)$ is the subgroup generated by all $p$-th powers, namely $\mho_k(G)=\langle g^{p^k}|g\in G\rangle$. Finally I call $G'$ the derived subgroup of $G$. My question is the following:
When is true that $\Phi(G)=G'\mho_1(G)$?
In general is true that $G'\mho_1(G) \le \Phi(G)$.
I found that equation reading the proof of Hall's Theorem for $p$-groups on Huppert's book "Endliche Gruppen I", at page 358. There $G$ is non cyclic with every abelian characteristic subgroup being cyclic. As it shown in the proof, this hypothesis implies that
- $\Phi(G)$ is cyclic
- $G/\Omega_1(G)$ is cyclic
- $G$ is an extraspecial (and then regular?) $p$-group
I don't find out why that equation seems to be kind of general.