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normal subgroup

Definition

Normal subgroups

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

A subgroup $N$ is normal iff the partition? of the group into left cosets of the subgroup $N$, that is the sets $gN=\{gn\mid n\in N\}$, 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 $gN=gN{g}^{-1}g=Ng$ 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).

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 $\mathrm{\infty }$-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 $BK\to BG$ (see example 3 below). The following definition takes this as the defining property of “normality” of morphisms.

Definition

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

A morphism $f:K\to G$ of ∞-groups in $H$ 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

$$BK\stackrel{Bf}{\to }BG\to B(G⫽K).$$\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 $G⫽K$. By the discussion at fiber sequence it follows that $G⫽K\simeq \Omega B(G⫽K)$ is the homotopy fiber of $Bf$, hence that we have a long fiber sequence

$$G⫽K\to BK\stackrel{Bf}{\to }BG\to BG/K.$$G\sslash K \to \mathbf{B}K \stackrel{\mathbf{B}f}{\to}\mathbf{B}G \to \mathbf{B}G/K \,.

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

For the case $H=$ ∞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 $G⫽K$ is a 2-group (example 3 below) is discussed in (Farjoun-Segev).

Remark

Such a normal morphism equivalently exhibits an ∞-group extension $G$ of $G⫽K$ 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 $\mathrm{\infty }$-groups. See example 3 below.

Properties

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 $\rho :G\to \mathrm{Aut}(N)$, the canonical inclusion

$$N\hookrightarrow G⋉N$$N \hookrightarrow G \ltimes N

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

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

$$G⫽K=(G\times K\stackrel{\stackrel{(-)\cdot f(-)}{\to }}{\underset{p_1}{\to }}G).$$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 $G⫽K$ 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).