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A subgroup is normal iff it is stable under conjugation by every group element
Normal Subgroup Criterion: Let be a group and let be a subgroup. Then is a normal subgroup of if and only if for every and every one has .
Equivalently, is normal if and only if for all (where ). This says precisely that is invariant under the conjugation action of on itself.
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 .
Let be a group. The conjugation action of on itself is the group action