Intersection of all maximal subgroups of a finite group

Question

Let $G$ be a group. Define $$F=\left\lbrace x\in G\;:\; \text{if } \left\langle S,x\right\rangle=G\text{ then }\left\langle S\right\rangle=G\right\rbrace.$$ I have that $F$ is a normal subgroup of $G$. In particular, if $G$ is finite, I have that $F$ is the intersection of all maximal subgroups of $G$. I have trouble showing that: If $G$ is a (finite) $p$-group ($p$ is a prime number), then every non-identity element of $G/F$ has order $p$.