A subgroup is normal iff the partition of the group into left cosets of the subgroup , that is the sets , is stable in the sense that the left coset of the product of any two elements depends only on the coset , . Thus there is well defined product on the set of cosets making the set of left cosets a group. By 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 is the same and called the quotient group (see quotient object).
Thus normal subgroups may also be defined as kernels of group homomorphisms ( is the identity), and this is largely the point of normal subgroups: they are equivalent to congruence relations in the category of groups.
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 -group, then a normal subgroup corresponds to an ideal?.
The notion of normal subgroups generalizes from groups to ∞-groups.
We may take as the characteristic propery of normal subgroup inclusions that the quotient inherits a group structure. This quotient may be identified with the homotopy fiber of the induced morphism of delooping groupoids (see example 3 below). The following definition takes this as the defining property of “normality” of morphisms.
Here the object on the right is any 0-connected ∞-groupoid. By the assumption of homotopy dimension 0 and by the discussion at looping and delooping this is necessarily the delooping of some ∞-group, to be denoted . By the discussion at fiber sequence it follows that is the homotopy fiber of , hence that we have a long fiber sequence
For the case ∞Grpd – hence for discrete ∞-groups – and with ∞Grpd presented by the standard model structure on topological spaces, this notion is discussed in (Prezma). The further special where is a morphism of discrete 1-groups, such that is a 2-group (example 3 below) is discussed in (Farjoun-Segev).
Such a normal morphism equivalently exhibits an ∞-group extension of by . See there for more details.
Every ordinary normal subgroup inclusion is also a normal morphism of ∞-groups, but there are more morphisms of 1-groups that are normal as morphisms of -groups. See example 3 below.
exhibits as a normal subgroup of the semidirect product group .
If is a normal subgroup of and is a group homomorphism, then the inverse image is normal in and induces a group homomorphism .
The proof is entirely straightforward and will be omitted.
This observation apparently goes back to Whitehead.
Normal morphisms of discrete ∞-groups are discussed in
The special case of this for morphisms of 1-groups is discussed in