Frattini's argument

If a subgroup is normal in a finite group, the group is generated by that subgroup and the normalizer of any Sylow subgroup of it.

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Sylow theorems: Theorems describing the existence, conjugacy, and number of Sylow subgroups in finite groups. Conjugacy of Sylow subgroups supplies the key step in the argument.
Normal subgroup: A subgroup invariant under conjugation by every element of its ambient group. Normality allows conjugates of the chosen Sylow subgroup to remain inside the subgroup.
Frattini subgroup: The intersection of all maximal subgroups of a finite group. Its normality lets Frattini's argument constrain normalizers of Sylow subgroups.
Giovanni Frattini: An Italian mathematician known for work on group theory and the Frattini subgroup. The argument and the Frattini subgroup carry his name.
Normalizer: For a subgroup H of G, the normalizer consists of elements g for which gHg⁻¹ equals H. The argument identifies each element of the group as a product involving this subgroup.
Finite group: A group with finitely many elements. The argument concerns finite groups, where Sylow subgroups exist.
Maximal subgroup: A proper subgroup contained in no larger proper subgroup. Applying the argument to a normal subgroup inside a maximal subgroup yields useful containment tests.
Frattini subgroup of a p-group: For a finite p-group, the Frattini subgroup is the intersection of its maximal subgroups. The argument's name is closely associated with this subgroup's role in generation.
Conjugacy class: The set of elements obtained by conjugating a given element or subgroup by every element of a group. Sylow conjugacy lets an arbitrary conjugate be matched by an element of the chosen normalizer.
Subgroup: A subset of a group that is itself a group under the same operation. The theorem relates a normal subgroup to a Sylow subgroup contained within it.