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In an abelian group every subgroup is normal
Proposition (Subgroups of abelian groups are normal). Let be an abelian group and let be a subgroup. Then is a normal subgroup of .
Context. Normality is a conjugation-invariance condition. In an abelian group, conjugation is trivial: for all .
An abelian group is a group such that for all ,
(That is, the group operation is commutative.)
Abelian groups are often written in additive notation, replacing by , the identity element by , and inverses by negatives. They form the algebraic backbone of linear structures (e.g. every vector space is an abelian group under addition).
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 .