nLab normal subgroup

Definition

Normal subgroups

A subgroup $N$ of a group $G$ is normal if the conjugation $n\mapsto g^{-1}n g$ by any element $g\in G$ leaves $N$ invariant, i.e. $g^{-1}N g \coloneqq \{g^{-1}n g\,|\,n\in N\} = N$.

A subgroup $N$ is normal iff the partition of the group into left cosets of the subgroup $N$, that is the sets $g N = \{ g n\,|\,n\in N\}$, is stable in the sense that the left coset $g_1 g_2 N$ of the product $g_1 g_2$ of any two elements $g_1,g_2\in G$ depends only on the coset $g_1 N$, $g_2 N$. Thus there is well defined product on the set of cosets making the set of left cosets $N\backslash G$ a group. By $g N = g N g^{-1}g = N g$ 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).

Thus normal subgroups may also be defined as kernels $f^{-1}(e)$ of group homomorphisms $G \to H$ ($e \in H$ is the identity), and this is largely the point of normal subgroups: they are equivalent to congruence relations in the category of groups.

Normal subobject in a semiabelian category

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 $\Omega$-group, then a normal subgroup corresponds to an ideal?.

Normal morphisms of $\infty$-groups

The notion of normal subgroups generalizes from groups to ∞-groups.

We may take as the characteristic propery of normal subgroup inclusions $K \hookrightarrow G$ that the quotient $G/K$ inherits a group structure. This quotient may be identified with the homotopy fiber of the induced morphism of delooping groupoids $\mathbf{B}K \to \mathbf{B}G$ (see example below). The following definition takes this as the defining property of “normality” of morphisms.

Definition

Let $\mathbf{H}$ be an (∞,1)-topos of homotopy dimension 0 (for instance a cohesive (∞,1)-topos) and let $K,G$ be ∞-group objects in $\mathbf{H}$.

A morphism $f : K \to G$ of ∞-groups in $\mathbf{H}$ is normal if its delooping is the homotopy fiber of a morphism to a 0-connected object, hence if it fits into a fiber sequence of the form

$$\mathbf{B}K \stackrel{\mathbf{B}f}{\to} \mathbf{B}G \to \mathbf{B}(G\sslash K) \,.$$

Remark

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 $G\sslash K$. By the discussion at fiber sequence it follows that $G\sslash K \simeq \Omega \mathbf{B}(G \sslash K)$ is the homotopy fiber of $\mathbf{B}f$, hence that we have a long fiber sequence

$$G\sslash K \to \mathbf{B}K \stackrel{\mathbf{B}f}{\to}\mathbf{B}G \to \mathbf{B}(G\sslash K) \,.$$

Therefore equivalently this says that $f : K \to G$ is normal precisely if $\mathbf{B}f : \mathbf{B}K \to \mathbf{B}G$ is a principal ∞-bundle. The above fiber sequence says that this principal $\infty$-bundle has typical fiber $G\sslash K$ and is classified by the cocycle $\mathbf{B}G \to \mathbf{B}(G\sslash K)$.

For the case $\mathbf{H} =$ ∞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 $f$ is a morphism of discrete 1-groups, such that $G\sslash K$ is a 2-group (example below) is discussed in (Farjoun-Segev).

Remark

Such a normal morphism equivalently exhibits an ∞-group extension $G$ of $G \sslash K$ by $K$. See there for more details.

Remark

Every ordinary normal subgroup inclusion $K \hookrightarrow G$ is also a normal morphism of ∞-groups, but there are more morphisms of 1-groups that are normal as morphisms of $\infty$-groups. See example below.

Properties

• The lattice of normal subgroups of a group $G$ is a modular lattice, because the category of groups is a Mal'cev category and, as mentioned earlier, normal subgroups are tantamount to congruence relations. (N.B.: it need not be true that the lattice of subgroups is modular: take for example the lattice of subgroups of the dihedral group of order $8$, which contains a forbidden pentagon.)

Recognition of homotopy-normal maps

A recognition principle for normality of morphisms of ∞-groups is (Prezma, theorem 6.2).

Examples

Normal sub-1-groups

Example

Every subgroup of an abelian group is normal, trivially.

Example

For $G$ a group equipped with an action on another group $N$ by group automorphisms $\rho : G \to Aut(N)$, the canonical inclusion

$$N \hookrightarrow G \ltimes N$$

exhibits $N$ as a normal subgroup of the semidirect product group $G \ltimes N$.

Proposition

If $N$ is a normal subgroup of $H$ and $\phi: G \to H$ is a group homomorphism, then the inverse image $\phi^{-1}(N)$ is normal in $G$ and $\phi$ induces a group homomorphism $G/f^{-1}(N) \to H/N$.

The proof is entirely straightforward and will be omitted.

Normal sub-2-groups

Example

Let $f : K \to G$ be a morphism of discrete groups (not necessarily a monomorphism) regarded as a morphisms of 0-truncated discrete ∞-groups. Then the homotopy fiber of its delooping is the action groupoid

$$G\sslash K = \left( G \times K \stackrel{\overset{(-)\cdot f(-)}{\to}}{\underset{p_1} {\to}} G \right) \,.$$

(This follows for instance by computing the homotopy pullback via the factorization lemma.)

Since $G\sslash K$ is a 1-type, this being an ∞-group means that it is a 2-group, hence (see the discussion there) that $f : K \to G$ makes a crossed module of groups.

So normal morphisms of 0-truncated discrete ∞-groups are equivalently morphisms underlying crossed modules of discrete groups.

This observation apparently goes back to Whitehead.

References

Normal morphisms of discrete ∞-groups are discussed in

• Matan Prezma, Homotopy Normal Maps (2010) (arXiv:1011.4708)

The special case of this for morphisms of 1-groups is discussed in

• E. D. Farjoun and Y. Segev, Crossed modules as homotopy normal maps, Topology and its applications 157 pp. 359–368 (2010).