Frattini subgroup

Remarks

An element $g \in G$ is a non-generator if, for every subset $X \subseteq G$ with $\langle X, g \rangle = G$, already $\langle X \rangle = G$. Then $\Phi(G)$ is the set of non-generators of $G$. $\Phi(G)$ is a characteristic subgroup of $G$, and if $G$ is finite it is nilpotent. For a finite $p$-group $P$, $\Phi(P)$ is generated by the $p$-th powers and the commutators, and $P/\Phi(P)$ is an elementary abelian group.

Example: $G = \Z/8\Z$. Its subgroups form a chain $\set{0} < \set{0, 4} < \set{0, 2, 4, 6} < G$, so the only maximal subgroup is $\set{0, 2, 4, 6}$, and this is $\Phi(G)$.

Example: $G = S_3$. The maximal subgroups are $A_3$ and the three subgroups of order $2$; their intersection is trivial, so $\Phi(S_3) = \set{e}$.

Example: the group $\Q$ of rational numbers under addition has no maximal subgroup, so $\Phi(\Q) = \Q$.

Non-example: in $\Z/8\Z$, the subgroup $\set{0, 4}$ is not the Frattini subgroup, since it is smaller than the intersection of the maximal subgroups.

Do not confuse $\Phi(G)$ with the Frattini argument, which is a different result named after the same mathematician.