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If N is normal and P is a Sylow p-subgroup of N, then G = N N_G(P)
Frattini Argument: Let be a finite group, let be a normal subgroup, and let be a Sylow p-subgroup of (so divides ). Let denote the normalizer of in . Then
Equivalently, every can be written as with and .
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
Let be a group and let be a subgroup. The subgroup is normal in (written ) if for every ,
Normality says that is stable under the conjugation action of on itself. Equivalently, is normal iff every left coset of equals the corresponding right coset, and this is exactly the hypothesis needed to form the quotient group .