# Intersection of all maximal subgroups of a finite group

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 $$p$$.

• math.stackexchange.com/questions/352331/… – Tsemo Aristide Dec 24 '18 at 0:54
• 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? – Hempelicious Dec 24 '18 at 1:40
• You should specify $S$ in your definition. And look at the concept of Frattini subgroups. – Nicky Hekster Dec 24 '18 at 8:11
• $S$ is a subset of $G$, you can deduce that from the definition. If $S$ and $x$ form a generator for $G$, then $S$ generates $G$ – José Alejandro Aburto Araneda Dec 24 '18 at 13:11