Suppose $G$ is a finite $p$-group. Let \begin{align*} \mho_{1}(G)=\langle a^p\mid a\in G\rangle,\quad\Omega_{1}(G)=\langle a\in G\mid a^p=1\rangle. \end{align*} There are examples such that $|G|\leq |\mho_{1}(G)||\Omega_{1}(G)|$ and $|G|\geq |\mho_{1}(G)||\Omega_{1}(G)|$. For $G=Q_{8}$, we computed $|\mho_{1}(G)||\Omega_{1}(G)|=4$. It's easy to construct $p$-groups of nilpotency class $p$ satisfy $|\mho_{1}(G)||\Omega_{1}(G)|$ strict less than $|G|$ for any odd prime number.
For $G$ be an arbitrary regular $p$-group, from P. Hall's known paper in 1932, we know $|G|= |\mho_{1}(G)||\Omega_{1}(G)|$.
We can get some $p$-groups satisfy $|G|$ strict less than $|\mho_{1}(G)||\Omega_{1}(G)|$ from $p$-central group. My question is that whether $|\mho_{1}(G)||\Omega_{1}(G)|$ is characterized by $|G|$ in general. I hope get lower bound and upper bound of $|\mho_{1}(G)||\Omega_{1}(G)|$ from $|G|$.
