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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nA *maximal subgroup* of $G$ is a [[subgroup|0X1JXT47GA]] $M \\ne G$ such that no subgroup $H$ satisfies $M \u003c H \u003c G$.\nThe *Frattini subgroup* $\\Phi(G)$ is the [[intersection|0839XNQXDK]] of all maximal subgroups of $G$, with $\\Phi(G) = G$ when $G$ has no maximal subgroup.\n\n## Remarks\n\nAn element $g \\in G$ is a *non-generator* if, for every subset $X \\subseteq G$ with $\\langle X, g \\rangle = G$, already $\\langle X \\rangle = G$.\nThen $\\Phi(G)$ is the set of non-generators of $G$.\n$\\Phi(G)$ is a [[characteristic subgroup|0KABPMRHM2]] of $G$, and if $G$ is finite it is [[nilpotent|0Q7MMQHC1P]].\nFor a finite [[$p$-group|06QYRTX9KQ]] $P$, $\\Phi(P)$ is generated by the $p$-th powers and the commutators, and $P/\\Phi(P)$ is an elementary abelian group.\n\nExample: $G = \\Z/8\\Z$.\nIts subgroups form a chain $\\set{0} \u003c \\set{0, 4} \u003c \\set{0, 2, 4, 6} \u003c G$, so the only maximal subgroup is $\\set{0, 2, 4, 6}$, and this is $\\Phi(G)$.\n\nExample: $G = S_3$.\nThe maximal subgroups are $A_3$ and the three subgroups of order $2$; their intersection is trivial, so $\\Phi(S_3) = \\set{e}$.\n\nExample: the group $\\Q$ of [[rational numbers|0NCJ5WXN6Z]] under addition has no maximal subgroup, so $\\Phi(\\Q) = \\Q$.\n\nNon-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.\n\nDo not confuse $\\Phi(G)$ with the [[Frattini argument|0QTZDKJEDV]], which is a different result named after the same mathematician.\n",
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