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  "body": "\n## Summary\n\nThe Frattini subgroup $\\Phi(G)$ of a group $G$ is the intersection of all maximal subgroups of $G$.\nIt is exactly the set of non-generators: elements that can be dropped from any generating set without changing the group generated.\nIt is a [[characteristic subgroup|0KABPMRHM2]], and for a finite group it is [[nilpotent|0Q7MMQHC1P]].\nFor a finite [[p-group|06QYRTX9KQ]] $P$, the quotient $P/\\Phi(P)$ is elementary abelian, and its dimension is the minimal number of generators of $P$.\n\n## Explanations\n\n### L1 human\n\nSome elements of a group are useless for generating it.\nIf a set of elements together with such an element generates the whole group, the set already generates it without that element.\nThe Frattini subgroup collects all these useless elements.\nIn the cyclic group $\\Z/8\\Z$, the even residues are useless: any generating set must contain an odd residue, which alone generates the group.\nSo the Frattini subgroup of $\\Z/8\\Z$ is $\\set{0, 2, 4, 6}$.\n",
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