nLab
normal subgroup

A subgroup N of a group G is normal if the conjugation ng 1ng by any element gG leaves N invariant, i.e. g 1Ng:={g 1ngnN}=N.

A subgroup N is normal iff the partition of the group into left cosets of the subgroup N, that is the sets gN={gnnN}, is stable in the sense that the left coset g 1g 2N of the product g 1g 2 of any two elements g 1,g 2G depends only on the coset g 1N, g 2N. Thus there is well defined product on the set of cosets making the set of left cosets N\G a group. By gN=gNg 1g=Ng the set of left cosets and the set of right cosets of a normal subgroup coincide; thus the induced group structure on the right coset set G/N is the same and called the quotient group? (see quotient object).

A normal subgroup is a normal subobject of a group in the category of groups: the more general notion of ‘normal subobject’ makes sense in semiabelian categories and some other setups. If we consider a group as a special case of an Ω-group, then a normal subgroup corresponds to an ideal?.

Of course, every subgroup of an abelian group is normal.