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normal subgroup

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Definition

Normal subgroups

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

A subgroup NN is normal iff the partition of the group into left cosets of the subgroup NN, that is the sets gN={gn|nN}g N = \{ g n\,|\,n\in N\}, is stable in the sense that the left coset g 1g 2Ng_1 g_2 N of the product g 1g 2g_1 g_2 of any two elements g 1,g 2Gg_1,g_2\in G depends only on the coset g 1Ng_1 N, g 2Ng_2 N. Thus there is well defined product on the set of cosets making the set of left cosets N\GN\backslash G a group. By gN=gNg 1g=Ngg 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/NG/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 Ω\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 KGK \hookrightarrow G that the quotient G/KG/K inherits a group structure. This quotient may be identified with the homotopy fiber of the induced morphism of delooping groupoids BKBG\mathbf{B}K \to \mathbf{B}G (see example 3 below). The following definition takes this as the defining property of “normality” of morphisms.

Definition

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

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

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

GKBKBfBGBG/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:KGf : K \to G is normal precisely if Bf:BKBG\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 GKG\sslash K and is classified by the cocycle BGB(GK)\mathbf{B}G \to \mathbf{B}(G\sslash K).

For the case H=\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 ff is a morphism of discrete 1-groups, such that GKG\sslash K is a 2-group (example 3 below) is discussed in (Farjoun-Segev).

Remark

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

Remark

Every ordinary normal subgroup inclusion KGK \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 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 GG a group equipped with an action on another group NN by group automorphisms ρ:GAut(N)\rho : G \to Aut(N), the canonical inclusion

NGN N \hookrightarrow G \ltimes N

exhibits NN as a normal subgroup of the semidirect product group GNG \ltimes N.

Normal sub-2-groups

Example

Let f:KGf : 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

GK=(G×Kp 1()f()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 GKG\sslash K is a 1-type, this being an ∞-group means that it is a 2-group, hence (see the discussion there) that f:KGf : 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

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).

Revised on August 30, 2013 17:45:33 by Urs Schreiber (89.204.154.236)