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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 $p$.

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  • $\begingroup$ math.stackexchange.com/questions/352331/… $\endgroup$ – Tsemo Aristide Dec 24 '18 at 0:54
  • $\begingroup$ You can show this directly: in a p-group (or any finite nilpotent group), every maximal subgroup is normal. What does that mean the quotient looks like? $\endgroup$ – Hempelicious Dec 24 '18 at 1:40
  • $\begingroup$ You should specify $S$ in your definition. And look at the concept of Frattini subgroups. $\endgroup$ – Nicky Hekster Dec 24 '18 at 8:11
  • $\begingroup$ $S$ is a subset of $G$, you can deduce that from the definition. If $S$ and $x$ form a generator for $G$, then $S$ generates $G$ $\endgroup$ – José Alejandro Aburto Araneda Dec 24 '18 at 13:11

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