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).
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