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group extension

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Idea

A group extension of a group GG by a group AA is third group G^\hat G that sits in a short exact sequence, that can usefully be thought of as a fiber sequence, AG^GA \to \hat G \to G.

Definition

Definition

Two consecutive homomorphisms of groups

(1)AiG^pG A \overset{i}\hookrightarrow \hat G\overset{p}\to G

are a short exact sequence if

  1. ii is monomorphism,

  2. pp an epimorphism

  3. the image of ii is all of the kernel of pp: ker(p)im(i)ker(p)\simeq im(i).

We say that such a short exact sequence exhibits G^\hat G as an extension of GG by AA.

If AG^A \hookrightarrow \hat G factors through the center of G^\hat G we say that this is a central extension.

Remark

Sometimes in the literature one sees G^\hat G called an extension “of AA by GG”. This is however in conflict with terms such as central extension, extension of principal bundles, etc, where the extension is always regarded of the base, by the fiber.

Under the looping and delooping-equivalence, this is equivalently reformulated as follows. For GG \in Grp a group, write BG\mathbf{B}G \in Grpd for its delooping groupoid.

Definition

A sequence AG^GA \to \hat G \to G is a short exact sequence of groups precisely if the delooping

BABG^BG \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G

is a fiber sequence in the (2,1)-category Grpd.

This says that group extensions are special cases of the general notion discussed at ∞-group extension. See there for more details.

Definition

A homomorphism of extensions f:G^ 1G^ 2f : \hat G_1 \to \hat G_2 of a given GG by a given AA is a group homomorphism of this form which fits into a commuting diagram

G^ 1 A f G G^ 2. \array{ && \hat G_1 \\ & \nearrow && \searrow \\ A &&\downarrow^{\mathrlap{f}}&& G \\ & \searrow && \nearrow \\ && \hat G_2 } \,.
Proposition

A morphism of extensions as in def. 3 is necessarily an isomorphism.

(2)1 A i G^ 1 p G 1 = ϵ = 1 A i G^ 2 p G 1. \array{ 1\to &A&\stackrel{i}\to &\hat G_1&\stackrel{p}\to &G&\to 1 \\ &\downarrow\mathrlap{=}&&\downarrow\mathrlap\epsilon&&\downarrow\mathrlap=& \\ 1\to &A&\stackrel{i'}\to &\hat G_2&\stackrel{p'}\to& G&\to 1 } \,.
Proof

By the short five lemma.

Properties

We discuss properties of group extensions in stages,

General

Fibers of extensions are normal subgroups

Proposition

For AG^GA \hookrightarrow \hat G \to G a group extension, the inclusion AG^A \hookrightarrow \hat G is a normal subgroup inclusion.

Proof

We need to check that for all aAGa \in A \hookrightarrow G and gGg \in G the result of the adjoint action gag 1g a g^{-1} formed in G^\hat G is again in AiG^A \stackrel{i}{\hookrightarrow} \hat G.

Since p:G^Gp : \hat G \to G is a group homomorphism we have that

p(gag 1) =p(q)p(a)p(g 1) =p(g)p(a)p(g) 1 =p(g)p(g) 1 =1 \begin{aligned} p(g a g^{-1}) & = p(q) p(a) p(g^{-1}) \\ & = p(g) p(a) p(g)^{-1} \\ & = p(g) p(g)^{-1} \\ & = 1 \end{aligned}

and hence gag 1g a g^{-1} is in the kernel of pp. By the defining exactness property therefore it is in the image of ii.

Extensions as torsors / principal bundles

Proposition

For AiG^pGA \stackrel{i}{\to} \hat G \stackrel{p}{\to} G a group extension, we have that p:G^Gp : \hat G \to G is an AA-torsor over GG where the action of AA on G^\hat G is defined by

ρ:A×G^(i,Id)G^×G^G^. \rho : A \times \hat G \stackrel{(i,Id)}{\to} \hat G \times \hat G \stackrel{\cdot}{\to} \hat G \,.
Proof

That ρ\rho is indeed an action over BB in that

A×G^ ρ G^ pp 2 p G^ \array{ A \times \hat G &&\stackrel{\rho}{\to}&& \hat G \\ & {}_{\mathllap{ p \circ p_2}}\searrow && \swarrow_{\mathrlap{p}} \\ && \hat G }

follows from the fact that pp is a group homomorphism and that AA is in its kernel.

That AA is actually equal to the kernel gives the principality condition

(ρ,p 2):A×G^G^× GG^. (\rho, p_2) : A\times \hat G \stackrel{\simeq}{\to} \hat G \times_G \hat G \,.

For AA an abelian group we may understand the AA-torsor/AA-principal bundle G^\hat G as the delooping of the BA\mathbf{B}A-principal 2-bundle BG^BG\mathbf{B} \hat G \to \mathbf{BG} that is classified by (is the homotopy fiber of) the 2-cocycle in group cohomology c:BGB 2Ac : \mathbf{B}G \to \mathbf{B}^2 A that classifies the extension.

All this is then summarized by the statement that

AG^GBABG^BGcB 2A A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \stackrel{c}{\to} \mathbf{B}^2 A

is a fiber sequence in ∞Grpd (or in ∞LieGrpd if we have Lie group extensions, etc).

Here we may think of BG^\mathbf{B}\hat G as being the BA\mathbf{B}A-principal 2-bundle over BG\mathbf{B}G classified by cc. See the examples discussed at bundle gerbe.

Split extensions and semidirect product groups

Definition

A group extension AG^pGA \to \hat G \stackrel{p}{\to} G is called split if there is a section homomorphism σ:GG^\sigma \colon G \to \hat G, hence a group homomorphism such that pσ=id Gp \circ \sigma = id_G.

Remark

It is important here that σ\sigma is itself required to be a group homomorphism, not just a function on the underlying sets. The latter always exists as soon as the axiom of choice holds, since pp is an epimorphism by definition.

Proposition

Split extensions G^\hat G of GG by AA, def. 4, are equivalently semidirect product groups

AG^A ρGG A \hookrightarrow \hat G \simeq A \rtimes_\rho G \to G

for some action ρ:G×AA\rho \colon G \times A \to A of GG on AA.

This means that the underlying set is U(A ρG)=U(A)×U(G)U(A \rtimes_\rho G) = U(A) \times U(G) and the group operation in A ρGA \rtimes_\rho G is

(a 1,g 1)(a 2,g 2)=(a 1ρ(g 1)(a 2),g 1g 2). (a_1, g_1) \cdot (a_2, g_2) = (a_1 \cdot \rho(g_1)(a_2) , g_1 \cdot g_2) \,.

The inclusion of AA is by

a(a,e) a \mapsto (a,e)

and the projection to GG is by

(a,g)g. (a,g) \mapsto g \,.
Proof

Given a split extension AiG^pGA \stackrel{i}{\to} \hat G \stackrel{p}{\to} G with splitting σ:GG^\sigma \colon G \to \hat G, define an action of GG on AA by the restriction of the adjoint action ρ ad\rho_{ad} of G^\hat G on itself to AA:

ρ:A×G(i,σ)G^×G^ρ adG^ \rho \colon A \times G \stackrel{(i,\sigma)}{\to} \hat G \times \hat G \stackrel{\rho_{ad}}{\to} \hat G
(a,g)σ(g) 1aσ(g). (a,g) \mapsto \sigma(g)^{-1} \cdot a \cdot \sigma(g) \,.

Then (…)

Proposition

A split extension AG^GA \to \hat G \to G is a central extension precisely if the action ρ\rho induced from it as in prop. 4 is trivial.

Proof

For it to be a central extension the inclusion AA ρGA \to A \rtimes_\rho G has to land in the center of A ρGA \rtimes_\rho G, hence all elements aAa \in A have to commute as elements (a,e)A ρG(a,e) \in A \rtimes_\rho G with all elements of A ρGA \rtimes_\rho G. But consider elements of the form (e,g)A ρG(e,g) \in A \rtimes_\rho G for all gGg \in G. Then

(a,e)(e,g)=(a,g) (a,e) \cdot (e,g) = (a, g)

but

(e,g)(a,e)=(ρ(g)(a),g). (e,g) \cdot (a,e) = (\rho(g)(a), g) \,.

For these to be equal for all aAa \in A, ρ(g)\rho(g) has to be the identity. Since this is to be true for all gGg \in G, the action has to be trivial.

Remark

This means in particular that split central extensions are product groups AGA \to G. If all groups involved are abelian groups, then these are equivalently the direct sums AGA \otimes G of abelian groups. In this way the notion of split group extension reduces to that of split short exact sequences of abelian groups.

Remark

If we have a split extension the different splittings are given by derivations, but with possibly non-abelian values. In fact if we have s:GAGs: G\to A\rtimes G is a section then s(g)=(a(g),g)s(g) = (a(g),g), and the multiplication in AGA\rtimes G implies that a:GAa: G\to A is a derivation. These are considered as the (possibly non-abelian) 1-cocylces of GG with (twisted) coefficients in AA, as considered in, for instance, Serre’s notes on Galois cohomology.

Central group extensions

We discuss properties of central group extensions, those where AG^A \hookrightarrow \hat G factors through the center of G^\hat G. This is a special case of the general discussion below in Nonabelian group extensions (Schreier theory) but is considerably less complex to write out in components.

We first discuss the

of central extensions in components, and then show in

how this follows from a more systematic abstract theory.

Classification by group cohomology

We discuss the classification of central extensions by group cohomology. This is a special case of the more general (and more complicated) discussion below in Nonabelian group extensions (Schreier theory).

For GG a group and AA an abelian group, write

H grp 2(G,A)Ab H^2_{grp}(G,A) \;\; \in Ab

for the degree-2 group cohomology of GG with coefficients in AA, and write

Ext(G,A)Ab Ext(G,A) \;\;\in Ab

for the group of central extensions of GG by AA.

Theorem

There is a natural equivalence

Ext(G,A)H Grp 2(G,A). Ext(G,A) \simeq H^2_{Grp}(G,A) \,.

We prove this below as prop. 10. Here we first introduce stepwise the ingredients that go into the proof.

Definition

(central extension associated to group 2-cocycle)

For [c]H Grp 2(G,A)[c] \in H^2_{Grp}(G,A) a group cohomology class represented by a cocycle c:G×GAc \colon G \times G \to A, define a group

G× cAGrp G \times_c A \in Grp

as follows. The underlying set is the cartesian product U(G)×U(A)U(G) \times U(A) of the underlying sets of GG and AA. The group operation on this is given by

(g 1,a 1)(g 2,a 2)(g 1g 2,a 1+a 2+c(g 1,g 2)). (g_1, a_1) \cdot (g_2, a_2) \coloneqq (g_1 \cdot g_2 ,\; a_1 + a_2 + c(g_1, g_2)) \,.
Proposition

This defines indeed a group: the cocycle condition on cc gives precisely the associativity of the product on G× cAG \times_c A. Moreover, the construction extends to a homomorphism of groups

Rec:H Grp 2(G,A)Ext(G,A). Rec : H^2_{Grp}(G,A) \to Ext(G,A) \,.
Proof

Forming the product of three elements of G× cAG \times_c A bracketed to the left is, according to def. 5,

((g 1,a 1)(g 2,a 2))(g 3,a 3)=(g 1g 2g 3,a 1+a 2+a 3+c(g 1,g 2)+c(g 1g 2,g 3)). \left( \left(g_1, a_1\right) \cdot \left(g_2, a_2\right) \right) \cdot \left( g_3, a_3 \right) = \left( g_1 g_2 g_3 \;,\; a_1 + a_2 + a_3 + c(g_1, g_2) + c\left( g_1 g_2, g_3 \right) \right) \,.

Bracketing the same three elements to the right yields

(g 1,a 1)((g 2,a 2)(g 3,a 3))=(g 1g 2g 3,a 1+a 2+a 3+c(g 2,g 3)+c(g 1,g 2g 3)). \left(g_1, a_1\right) \cdot \left( \left(g_2, a_2\right) \cdot \left( g_3, a_3 \right) \right) = \left( g_1 g_2 g_3 \;,\; a_1 + a_2 + a_3 + c(g_2, g_3) + c\left( g_1 , g_2 g_3 \right) \right) \,.

The difference between the two expressions is read off to be precisely

(1,(dc)(g 1,g 2,g 3)), (1, (d c) (g_1, g_2, g_3)) \,,

where dcd c denotes the group cohomology differential of cc. Hence this vanishes precisely if cc is a group 2-cocycle, hence we have an associative product.

To see that it has inverses, notice that for all (g,a)(g,a) we have

(g,a)(g 1,ac(g,g 1))=(e,aac(g,g 1)+c(g,g 1)) (g,a) \cdot (g^{-1}, - a - c(g,g^{-1})) = (e, a - a - c(g,g^{-1})+ c(g,g^{-1}) )

and hence inverses are given by (g,a) 1=(g 1,ac(g,g 1))(g,a)^{-1} = (g^{-1}, -a - c(g,g^{-1})). Hence G× cAG \times_c A is indeed a group.

By the discussion at group cohomology – degree-2 we may assume without restriction that cc is a normalized cocycle, hence that c(e,)=c(,e)=0c(e,-) = c(-,e) = 0. Using this we find that the inclusion

i:AG× cA i \colon A \to G \times_c A

given by a(e,a)a \mapsto (e,a) is a group homomorphism. Moreover, the projection on the underlying sets evidently yields a group homomorphism p:G× cAGp \colon G \times_c A \to G given by (g,a)g(g,a) \mapsto g. The kernel of this is AA, and hence

AiG× cApG A \stackrel{i}{\hookrightarrow} G \times_c A \stackrel{p}{\to} G

is indeed a group extension. It is a central extension again using the assumption that cc is normalized c(g,e)=c(e,g)=0c(g,e) = c(e,g) = 0:

(g,a)(e,a˜)=(g,a+a˜+0)=(e,a˜)(g,a). (g,a) \cdot (e,\tilde a) = (g, a + \tilde a + 0) = (e,\tilde a) \cdot (g,a) \,.

Finally to see that the construction is independent of the choice of coycle cc representing [c][c], let c˜\tilde c be another representative which differs by a coboundary h:GAh \colon G \to A with

c˜(g 1,g 2)c(g 1,g 2)h(g 1)h(g 2)+h(g 1g 2). \tilde c (g_1,g_2) \coloneqq c(g_1,g_2) - h(g_1) - h(g_2) + h(g_1 g_2) \,.

We claim that then we have a homomorphism of central extensions (hence an isomorphism) of the form

A G× cA G id (id G,p 2hp 1) = A G× c˜A G. \array{ A &\to& G \times_c A &\to& G \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{(id_G, p_2 -h \circ p_1)}} && \downarrow^{\mathrlap{=}} \\ A &\to& G \times_{\tilde c} A &\to& G } \,.

To see this we check for all elements that

(g 1,a 1h(g 1))(g 2,a 2h(g 2)) =(g 1g 2,a 1+a 2h(g 1)h(g 2)+c(g 1,g 2)) =(g 1g 2,a 1+a 2+c˜(g 1,g 2)h(g 1g 2)). \begin{aligned} (g_1, a_1 - h(g_1)) \cdot (g_2, a_2 - h(g_2)) & = (g_1 g_2, a_1 + a_2 - h(g_1) - h(g_2) + c(g_1, g_2)) \\ & = (g_1 g_2, a_1 + a_2 + \tilde c(g_1, g_2) - h(g_1 g_2) ) \end{aligned} \,.

Hence the construction of G× cAG \times_c A indeed defines a function H Grp 2(G,A)CentrExt(G,A)H^2_{Grp}(G,A) \to CentrExt(G,A).

Assume the axiom of choice in the ambient foundations.

Definition

(2-cocycle extracted from central extension)

Given a central extension AG^GA \to \hat G \to G define a group 2-cocycle c:G×GAc : G \times G \to A as follows.

Choose a section σ:U(G)U(G^)\sigma : U(G) \to U(\hat G) of the underlying sets (which exists by the axiom of choice and the fact that p:G^Gp : \hat G \to G is by definition an epimorphism). Then define cc by

c:(g 1,g 2)σ(g 1) 1σ(g 2) 1σ(g 1g 2)A, c \colon (g_1, g_2) \mapsto -\sigma(g_1)^{-1} \sigma(g_2)^{-1} \sigma(g_1 g_2) \in A \,,

where on the right we are using that by the section-property of σ\sigma and the group homomorphism property of pp

p(σ(g 1) 1σ(g 2) 1σ(g 1g 2))=1 p(\sigma(g_1)^{-1} \sigma(g_2)^{-1} \sigma(g_1 g_2)) = 1

and hence by the exactness of the extension the argument is in AG^A \hookrightarrow \hat G.

Below in remark 3 is a discussion of how this construction arises from a more systematic discussion in homotopy theory.

Proposition

The construction of prop. 6 indeed yields a 2-cocycle in group cohomology. It extends to a morphism

Extr:Ext(G,A)H Grp 2(G,A). Extr \colon Ext(G,A) \to H^2_{Grp}(G,A) \,.
Proof

The cocycle condition to be checked is that

c(g 1,g 2)c(g 0g 1,g 2)+c(g 0,g 1g 2)c(g 0,g 1)=1 c(g_1, g_2) - c(g_0 g_1, g_2) + c(g_0, g_1 g_2) - c(g_0, g_1) = 1

for all g 0,g 1,g 2Gg_0, g_1, g_2 \in G. Writing this out with def. 6 yields

σ(g 1) 1σ(g 2) 1σ(g 1g 2)(σ(g 0g 1) 1σ(g 2) 1σ(g 0g 1g 2)) 1σ(g 0) 1σ(g 1g 2) 1σ(g 0g 1g 2)(σ(g 0) 1σ(g 1) 1σ(g 0g 1)) 1. \sigma(g_1)^{-1} \sigma(g_2)^{-1} \sigma(g_1 g_2) \left(\sigma(g_0 g_1)^{-1} \sigma(g_2)^{-1} \sigma(g_0 g_1 g_2)\right)^{-1} \sigma(g_0)^{-1} \sigma(g_1 g_2)^{-1} \sigma(g_0 g_1 g_2) \left( \sigma(g_0)^{-1} \sigma(g_1)^{-1} \sigma(g_0 g_1) \right)^{-1} \,.

Here it is sufficient to observe that for every term also the inverse term appears.

To see that this is a well-defined map to H grp 2(G,A)H^2_{grp}(G,A) we need to check that for σ˜:GG^\tilde \sigma : G \to \hat G a different choice of section, the corresponding cocycles differ by a group coboundary c˜c=dh\tilde c - c = d h. Clearly this is obtained by setting

h:gσ˜(g)σ(g) 1, h \colon g \mapsto \tilde \sigma(g)\sigma(g)^{-1} \,,

where we use that the right hand side is in AG^A \hookrightarrow \hat G since because both σ\sigma and σ˜\tilde \sigma are sections of pp, the image of the right hand under pp is the neutral element in GG.

Proposition

The two morphisms of def. 5 and def. 6 exhibit the equivalence

H Grp 2(G,A)RecExtrCentrExt(G,A). H^2_{Grp}(G,A) \stackrel{\underoverset{\simeq}{Extr}{\leftarrow}}{\underset{Rec}{\to}} CentrExt(G,A) \,.
Proof

Let [c]H Grp 2(G,A)[c] \in H^2_{Grp}(G,A). Then by construction of G^G× cA\hat G \coloneqq G \times_c A there is a canonical section of the underlying function of sets U(G× cA)U(G)U(G \times_c A) \to U(G) given by (id U(G),0)U(G)U(G)×U(A)(id_{U(G)}, 0) U(G) \to U(G) \times U(A). The cocycle induced by this section sends

(g 1,g 2) (g 1,0)(g 2,0)(g 1g 2,0) 1 =(g 1,0)(g 1,0)((g 1g 2) 1,c(g 1g 2,(g 1g 2) 1)) =(g 1g 2,c(g 1,g 2))((g 1g 2) 1,c(g 1g 2,(g 1g 2) 1)) =(e,c(g 1,g 2)c(g 1g 2,(g 1g 2) 1)+c(g 1g 2,(g 1g 2) 1)) =(e,c(g 1,g 2)), \begin{aligned} (g_1, g_2) & \mapsto (g_1, 0) (g_2, 0) (g_1 g_2, 0)^{-1} \\ & = (g_1, 0) (g_1, 0) ((g_1 g_2)^{-1}, - c(g_1 g_2, (g_1 g_2)^{-1}) ) \\ & = (g_1 g_2, c(g_1, g_2) ) ((g_1 g_2)^{-1}, - c(g_1 g_2, (g_1 g_2)^{-1}) ) \\ & = (e, c(g_1, g_2) - c(g_1 g_2, (g_1 g_2)^{-1}) + c(g_1 g_2, (g_1 g_2)^{-1})) \\ & = (e, c(g_1, g_2)) \end{aligned} \,,

which is c(g 1,g 2)AG× cAc(g_1, g_2) \in A \hookrightarrow G \times_c A, and hence this recovers the 2-cocycle that we started with.

This shows that ExtrRec=idExtr \circ Rec = id and in particular that RecRec is a surjection. It is readily seen that the kernel of RecRec is trivial, and so it is an equivalence.

Formulation in homotopy theory

We discuss the classification of central group extensions by degree-2 group cohomology in the more abstract context of homotopy theory, complementing the above component-wise discussion.

Let

AG^G A \hookrightarrow \hat G \to G

be a central group extension, def. 1, hence with AA an abelian group included in the center of GG. Then AA is in particular a normal subgroup and hence the homorphism

(AG^) (A \to \hat G)

may be regarded as a crossed module of groups. This is equivalently a strict 2-group structure on the groupoid whose objects are G^\hat G and whose morphisms are labeled in AA

(AG^)={gaag}. (A \to \hat G) = \{ g \stackrel{a}{\to} a \cdot g \} \,.
Definition

Write

B(AG^)Grpd \mathbf{B}(A \to \hat G) \in Grpd

for the delooping of this 2-group to a one-object 2-groupoid.

The ω-nerve (or Duskin nerve) NB(AG^)N \mathbf{B}(A \to \hat G) \in sSet of this is a (3-coskeletal) Kan complex that realizes this as a 2-truncated ∞-groupoid.

B(AG^)Grpd. \mathbf{B}(A \to \hat G) \in \infty Grpd \,.
Proposition

The obvious strict 2-functor

B(AG^)BH \mathbf{B}(A \to \hat G) \stackrel{\simeq}{\to} \mathbf{B}H

is an equivalence of 2-groupoids.

Proof

One way to see this is to notice that this is a k-surjective functor for all k{0,1,2,3}k \in \{0,1,2,3\}, hence a weak equivalence in the folk model structure on ω\omega-groupoids. Equivalently, under the nerve the morphism of simplicial sets

NB(AG^)NBH N\mathbf{B}(A \to \hat G) \to N \mathbf{B}H

is an acyclic Kan fibration, hence a weak equivalence in the standard model structure on simplicial sets.

Proposition

The extension AG^GA \to \hat G \to G sits in a long homotopy fiber sequence in the (∞,1)-category ∞Grpd of the form

AG^GBABG^BGcB 2A A \to \hat G \to G \to \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^2 A

which in Kan complexes/simplicial sets is presented by the zigzag of n-functors between strict ω-groupoid (sequence of 2-anafunctors) of the form

B(AG^) B(A1)=B 2A (AG^) BA BG^ BG A G^ G. \array{ && && && && && \mathbf{B}(A \to \hat G) &\to & \mathbf{B}(A \to 1) = \mathbf{B}^2 A \\ && && && && && {}^{\mathrlap{\simeq}}\downarrow \\ && && (A \to \hat G) &\to& \mathbf{B}A &\to& \mathbf{B}\hat G &\to& \mathbf{B}G \\ && && \downarrow^{\mathrlap{\simeq}} \\ A &\to& \hat G &\to& G } \,.

In particular, the induced connecting homomorphism

c:BGB 2A \mathbf{c} : \mathbf{B}G \to \mathbf{B}^2 A

is the group cohomology cocycle that classifies the delooped extension as a BA\mathbf{B}A-principal 2-bundle.

Proof

One sees directly that the morphisms BG^BG\mathbf{B}\hat G \to \mathbf{B}G and B(AG^)B 2A\mathbf{B}(A \to \hat G ) \to \mathbf{B}^2 A as well as their loopings G^G\hat G \to G and (AG^)G(A \to \hat G) \to G are Kan fibrations. By the discussion at homotopy pullback this means that the set-theoretic fibers of these morphisms are models for their homotopy fibers. But the ordinary kernel of B(AG^)B(A1)=B 2A\mathbf{B}(A \to \hat G) \to \mathbf{B}(A \to 1) = \mathbf{B}^2 A is manifestly BG^\mathbf{B} \hat G, and so on.

Remark

The construction in def. 6

Rec:CentrExt(G,A)H Grp 2(G,A) Rec : CentrExt(G,A) \to H^2_{Grp}(G,A)

is precisely the result of moving set-theoretically through the zigzag

B(AG^) B(A1)=B 2A BG \array{ \mathbf{B}(A \to \hat G) &\to& \mathbf{B}(A \to 1) = \mathbf{B}^2 A \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}G }

from the bottom left to the top right, and that this is well-defined on cohomology comes down to the statement that the vertical morphism is a weak homotopy equivalence.

This is a nonabelian analog of the discussion at mapping cone in the section Homology exact sequences and fiber sequences.

Abelian group extensions

Remark

For A,GA, G \in Ab \hookrightarrow Grp even a central extension G^\hat G of GG by AA is not necessarily itself an abelian group.

But by prop. 10 above it is so if the group 2-cocycle that classifies the extension is symmetric:

Definition

A 2-cocycle c:G×GAc \colon G \times G \to A in group cohomology is symmetric if

g 1,g 2Gc(g 1,g 2)=c(g 2,g 1). \forall_{g_1, g_2 \in G} c(g_1, g_2) = c(g_2, g_1) \,.
Remark

A group 2-cocycle cohomologous to a symmetric group 2-cocycle is itself symmetric. Hence we may speak of symmetric group cohomology classes in degree 2.

Definition

Write

H Grp 2(G,A) symH Grp 2(G,A) H^2_{Grp}(G,A)_{sym} \hookrightarrow H^2_{Grp}(G,A)

for the set (group) of classes of symmetric group 2-cocycles on GG with coefficients in AA.

Definition

For G,AAbGrpG,A \in Ab \hookrightarrow Grp, write Ext(G,A)Ext(G,A) for the subset of equivalence class of abelian group extensions of GG by AA.

The theory of abelian group extensions in Ab is naturally and classically treated with tools of homological algebra, such as the theory of Ext-functors.

For the moment see at projective resolution the section

and

Nonabelian group extensions (Schreier theory)

We discuss the classification theory for the general case of nonabelian group extensions, first in the form of

and then more abstractly in the language of homotopy theory in

Traditional description

Otto Schreier (1926) and Eilenberg-Mac Lane (late 1940-s) developed a theory of classification of nonabelian extensions of abstract groups leading to the low dimensional nonabelian group cohomology. This is sometimes called Schreier’s theory of nonabelian group extensions.

The traditional Schreier-Mac Lane way to obtain nonabelian group 2-cocycle from a group extension as above starts with choosing a set-theoretic section of p:GBp:G\to B.

Note. The exposition which follows in this long “traditional” section of this entry is mainly from personal notes of Zoran Škoda from 1997.

Each element gg of GG defines an inner automorphism ϕ(g)\phi(g) of KK by ϕ(g)(k)=gkg 1\phi(g)(k) = gkg^{-1}. The restriction ϕ K\phi|_K takes (by definition) values in the subgroup Int(K)Int(K) of inner automorphisms of KK. In fact ϕ:GInn(G)Aut(K)\phi:G\to Inn(G)\subset Aut(K) is a homomorphism of groups.

If g 1g_1 and g 2g_2 are in the same left coset, that is g 1K=g 2Kg_1K = g_2K, then there is kKk \in K, g 1=g 2kg_1 = g_2k, so that kK\forall k' \in K we have ϕ(g 1k)=ϕ(g 2kk)=ϕ(g 2)ϕ(kk)\phi(g_1k') = \phi(g_2kk') = \phi(g_2)\phi(kk') and therefore ϕ(g 1K)ϕ(g 2)Int(K)\phi(g_1K) \subset \phi(g_2)Int(K). Thus we obtain a well-defined map ϕ *:G/KAut(K)/Int(K)\phi_* : G/K \rightarrow Aut(K)/Int(K). Choose a set-theoretic section of the projection p:GBp : G \rightarrow B and let

(3)ψ=defϕσ:BAut(K).\psi \stackrel{def}{=} \phi \circ \sigma: B \rightarrow Aut(K).

Warning. Unlike ϕ\phi, the map ψ\psi is not a homomorphism of groups.

We attempt to reconstruct GG from the knowledge of ψ\psi and KK. As a set, GG can be naturally identified with B×KB \times K. Indeed, write each element gGg \in G as σ(b)k,bB,kK\sigma(b)k, b \in B, k \in K by setting b=p(g),k=σ(p(g)) 1gb = p(g), k = \sigma(p(g))^{-1}g. Elements bBb \in B and kKk \in K in that decomposition are unique, and we get a bijection

(4)B×K(b,k)σ(b)kG, B\times K\ni (b,k)\mapsto\sigma(b)k \in G,

whose inverse is the map g(p(g),σ(p(g)) 1g)g \mapsto (p(g), \sigma(p(g))^{-1}g). By means of that bijection, B×KB \times K inherits the group structure from GG. Let us figure out the multiplication rule on B×K.B \times K. If σ(b 1)k 1=g 1\sigma(b_1)k_1 = g_1, and σ(b 2)k 2=g 2\sigma(b_2)k_2 = g_2, then,

(5)g 1g 2=σ(b 1)k 1σ(b 2)k 2=σ(b 1)σ(b 2)σ(b 2) 1k 1σ(b 2)k 2. g_1g_2 = \sigma(b_1)k_1\sigma(b_2)k_2 = \sigma(b_1)\sigma(b_2)\sigma(b_2)^{-1}k_1\sigma(b_2)k_2.

Now p(σ(b 1)σ(b 2))=p(b 1b 2)p(\sigma(b_1)\sigma(b_2)) = p(b_1b_2) so

χ(b 1,b 2)=defσ(b 1b 2) 1σ(b 1)σ(b 2)K.\chi(b_1,b_2) \stackrel{def}{=} \sigma(b_1b_2)^{-1}\sigma(b_1)\sigma(b_2) \in K.

This formula clearly defines a function χ:B×BK\chi : B \times B \rightarrow K. In this notation,

g 1g 2 = σ(b 1b 2)χ(b 1,b 2)ϕ(σ(b 2) 1)(k 1)k 2 = σ(b 1b 2)χ(b 1,b 2)[ψ(b 2) 1(k 1)]k 2.\array{ g_1g_2 & = & \sigma(b_1b_2)\chi(b_1,b_2)\phi(\sigma(b_2)^{-1})(k_1)k_2 \\ & = & \sigma(b_1b_2)\chi(b_1,b_2)[\psi(b_2)^{-1}(k_1)] k_2. }

and using bijection of GG with B×KB\times K this can be expressed in terms of elements in B×KB\times K so that

(6)(b 1,k 1)(b 2,k 2)=(b 1b 2,χ(b 1,b 2)[ψ(b 2) 1(k 1)]k 2). (b_1,k_1)(b_2,k_2) = (b_1b_2,\chi(b_1,b_2)[\psi(b_2)^{-1}(k_1)] k_2).

According to this formula, all the information about the multiplication is encoded in functions χ:B×BAut(K)\chi : B \times B \rightarrow Aut(K) and ψ:BAut(K)\psi : B \rightarrow Aut(K), and we may forget about σ\sigma at this point. However, not every pair (χ,ψ)(\chi,\psi) will give some multiplication rule on B×KB \times K. Let a,b,cBa,b,c \in B, and e=e Ke = e_K be the unity element in KK. Then

[(a,e)(b,e)](c,e)=(ab,χ(a,b))(c,e)=(abc,χ(ab,c)ψ(c) 1(χ(a,b))). [(a,e)(b,e)](c,e) = (a b, \chi(a,b))(c,e) = (a b c, \chi(a b,c) \psi(c)^{-1}(\chi(a,b))).

From the other side, this has to be the same, by associativity, to

(a,e)[(b,e)(c,e)]=(a,e)(bc,χ(b,c))=(abc,χ(a,bc)χ(b,c)) (a,e)[(b,e)(c,e)] = (a,e)(bc,\chi(b,c)) = (a b c,\chi(a,b c)\chi(b,c))

where we took into account that expressions like [ψ 1(b)(e)]=e[\psi^{-1}(b)(e)] = e, because ψ(b)\psi(b) is an automorphism for each bBb \in B.

Comparing the expressions above we obtain

(7)χ(ab,c)ψ(c) 1(χ(a,b))=χ(a,bc)χ(b,c),foralla,b,cB. \chi(a b,c)\psi(c)^{-1}(\chi(a,b)) = \chi(a,b c)\chi(b,c), for all a,b,c \in B.

If the pair (χ,ψ)(\chi,\psi) is constructed as above, then

ψ(a)ψ(b)k = ϕ(σ(a))ϕ(σ(b))k = σ(a)σ(b)kσ(b) 1σ(a) 1 = σ(ab)χ(a,b)kχ(a,b) 1σ(ab) 1 = ψ(ab)Ad K(χ(a,b))k,\array{ \psi(a)\psi(b)k & = & \phi(\sigma(a))\phi(\sigma(b))k \\ & = & \sigma(a)\sigma(b)k\sigma(b)^{-1}\sigma(a)^{-1} \\ & = & \sigma(a b)\chi(a,b)k\chi(a,b)^{-1}\sigma(a b)^{-1} \\ & = & \psi(a b) Ad_K(\chi(a,b)) k, }

where Ad KAd_K is the canonical map KInt(K)K \rightarrow Int(K), kk()k 1k\mapsto k(-)k^{-1}.

Thus we obtain the relation

(8)ψ(a)ψ(b)=ψ(ab)Ad K(χ(a,b)) \psi(a)\psi(b) = \psi(a b) Ad_K(\chi(a,b))
Definition

Let BB and KK be two groups. Let χ:B×BK\chi: B \times B \rightarrow K and ψ:BAut(K)\psi : B \rightarrow Aut(K) satisfy (7) and (8). Then we call that the family {χ(b 1,b 2)b 1,b 2B}\{\chi(b_1,b_2)| b_1,b_2 \in B\} is a factor system (This term is due Schreier(1924)) or a nonabelian group 2-cocycle with automorphisms, and the family {ψ(b)bB}\{\psi(b) | b \in B \} – a system of automorphisms

A 2-cocycle χ\chi is counital if χ(b,e)=χ(e,b)=e\chi(b,e) = \chi(e,b) = e, for all bBb \in B.

If KK is commutative, then ψ\psi is always a homomorphism (cf. (8)). Then KK is a right BB-module through ψ() 1\psi(-)^{-1}. That justifies the sometimes used term “(right) cocycle BB-module” for (K,ψ,χ)(K,\psi,\chi). If ψ\psi is trivial (ψ(b)=Id K,bB\psi(b) = Id_K, \forall b \in B) then the cocycle condition (7) becomes

(9)χ(ab,c)χ(a,b)=χ(a,bc)χ(b,c). \chi(a b,c)\chi(a,b) = \chi(a,b c)\chi(b,c).
Theorem

If formulas (7) and (8) are both satisfied, then the formula (6) for multiplication of pairs defines a group multiplication on B×KB \times K. That set, together with multiplication (6) is called the cocycle cross product of BB and KK with cocycle χ\chi and action ψ\psi. If the cocycle is trivial i.e. χ(,)=e K\chi(\cdot,\cdot) = e_K, we call it the (external) semidirect product.

Proof

We have checked above the associativity for pairs of the form (a,e)(a,e) etc. This was useful to find the cocycle condition correctly. Now the general associativity should be a similar calculation with general elements. Using (7) and (8) it can be done.

ψ(a)ψ(e)k = ψ(a)Ad K(χ(a,e))k = ψ(a)χ(a,e)kχ(a,e) 1\array{ \psi(a)\psi(e)k & =& \psi(a)Ad_K(\chi(a,e))k \\ & = &\psi(a)\chi(a,e)k\chi(a,e)^{-1} }

where we used (8).

Thus Ad K(χ(a,e))=ψ(e)Ad_K(\chi(a,e)) = \psi(e) and therefore it does not depend on aa.

Then use (7) with b=c=eb = c = e to get ψ(e) 1(χ(a,e))=χ(e,e),aB\psi(e)^{-1}(\chi(a,e)) = \chi(e,e), \forall a \in B.

Thus χ(a,e) 1(χ(a,e))χ(a,e)=χ(e,e)\chi(a,e)^{-1}(\chi(a,e))\chi(a,e) = \chi(e,e), that is χ(a,e)\chi(a,e) does not depend on aa.

Now we claim that the unit element is given by (e,χ(e,e) 1)(e, \chi(e,e)^{-1}). To verify that it is also a right unit we compute

(a,b)(e,χ(e,e) 1) = (a,χ(a,e)ψ(e) 1(b)χ(e,e) 1) = (a,χ(a,e)χ(e,e) 1bχ(e,e)χ(e,e) 1)\array{ (a,b)(e,\chi(e,e)^{-1}) & = & (a, \chi(a,e) \psi(e)^{-1}(b)\chi(e,e)^{-1}) \\ & = & (a, \chi(a,e)\chi(e,e)^{-1}b\chi(e,e)\chi(e,e)^{-1}) }

what is equal to (a,b)(a,b) by just proved statement that χ(a,e)\chi(a,e) does not depend on aa.

Now use (7) with a=b=ea = b = e to get

(10)ψ(c) 1(χ(e,e))=χ(e,c),cB. \psi(c)^{-1}(\chi(e,e)) = \chi(e,c), \forall c \in B.

Thus we can verify that (e,χ(e,e) 1)(e, \chi(e,e)^{-1}) is a left unit too by a calculation as follows. Namely

(e,χ(e,e) 1)(a,b)=(a,χ(e,a)ψ(a) 1(χ(e,e) 1)b)(e,\chi(e,e)^{-1})(a,b)= (a,\chi(e,a)\psi(a)^{-1}(\chi(e,e)^{-1})b)

by the definition of the product. Then by (10), this equals to

(a,ψ(a) 1(χ(e,e))ψ(a) 1(χ(e,e) 1)b)(a,\psi(a)^{-1}(\chi(e,e))\psi(a)^{-1}(\chi(e,e)^{-1})b)

and, because ψ(a) 1\psi(a)^{-1} is an antiautomorphism, this is finally equal to (a,b)(a,b).

Now check that each element (a,b)(a,b) can be factorized as (a,e)(e,χ(e,e) 1b)(a,e)(e,\chi(e,e)^{-1}b). In order to show that (a,b)(a,b) has an inverse it is then enough to show that both (a,e)(a,e) and (e,χ(e,e) 1b)(e,\chi(e,e)^{-1}b) have inverses.

Claim: the inverse of (a,e)(a,e) is

(a 1,χ(a,a) 1χ(e,e) 1).(a^{-1},\chi(a,a)^{-1}\chi(e,e)^{-1}).

To this aim, we calculate

(a,e)(a 1,χ(a,a 1) 1χ(e,e) 1)=(e,χ(a,a 1)ψ(a 1) 1(e)χ(a,a 1) 1χ(e,e) 1)=(e,χ(e,e 1),(a,e)(a^{-1},\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) = (e,\chi(a,a^{-1})\psi(a^{-1})^{-1}(e)\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) =(e,\chi(e,e^{-1}),

because ψ(a) 1(e)=e\psi(a)^{-1}(e) = e. Furthermore,

(a,e)(a 1,χ(a,a 1) 1χ(e,e) 1)=(e,χ(a,a 1)ψ(a 1) 1(e)χ(a,a 1) 1χ(e,e) 1)=(e,χ(e,e 1),(a,e)(a^{-1},\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) = (e,\chi(a,a^{-1})\psi(a^{-1})^{-1}(e)\chi(a,a^{-1})^{-1}\chi(e,e)^{-1}) = (e,\chi(e,e^{-1}),

because ψ(a) 1(e)=e\psi(a)^{-1}(e) = e. Next,

(a 1,χ(a,a 1) 1χ(e,e) 1)(a,e)=(e,χ(a 1,a)ψ(a) 1(χ(a,a 1)χ(e,e) 1))(a^{-1},\chi(a,a^{-1})^{-1}\chi(e,e)^{-1})(a,e) = (e,\chi(a^{-1},a)\psi(a)^{-1}(\chi(a,a^{-1}) \chi(e,e)^{-1}))

what equals (e,χ(e,e) 1)(e,\chi(e,e)^{-1}).

Indeed, (7) with a=a,b=a 1,c=aa = a, b = a^{-1}, c = a reads χ(e,a)ψ(a) 1(χ(a,a 1))\chi(e,a) \psi(a)^{-1}(\chi(a, a^{-1})) =χ(a,e)χ(a 1,a)=\chi(a,e)\chi(a^{-1},a).

Then apply (10) and take inverse of both sides to obtain

ψ(a) 1(χ(a,a 1) 1χ(e,e) 1))=χ(a 1,a) 1χ(a,e) 1.\psi(a)^{-1}(\chi(a,a^{-1})^{-1}\chi(e,e)^{-1})) = \chi(a^{-1},a)^{-1}\chi(a,e)^{-1}.

Then recall that χ(a,e)\chi(a,e) does not depend on aa and multiply by χ(a 1,a)\chi(a^{-1},a) from the left.

Claim: the inverse of (e,χ(e,e) 1k)(e,\chi(e,e)^{-1}k) is (e,χ(e,e)k 1)(e,\chi(e,e)k^{-1}). Here the verification is symmetric (kk vs. k 1k^{-1}) for the left and for the right inverse and immediate.

Given groups KK and BB and any maps χ\chi and ψ\psi satisfying (7) and (8), needed to define a cocycle cross product B× χKB\times_\chi K of KK and BB, one defines the map i:KB× χKi : K \rightarrow B \times_\chi K by k(e,χ(e,e) 1k)k \mapsto (e,\chi(e,e)^{-1}k). Then ii is a monomorphism of groups, i(K)i(K) is a normal subroup of the cocycle cross product of BB and KK, and there is a canonical isomorphism BG/KB \cong G/K. We define the set-theoretic maps σ,χ\sigma',\chi' and ψ\psi' as follows. σ:BB×K\sigma' : B \rightarrow B \times K is defined by σ(b)=(b,e)\sigma'(b) = (b, e) , for all bBb \in B. Then χ:B×Bi(K)\chi' : B \times B \to i(K) is defined by χ(b 1,b 2)=σ(b 1b 2) 1σ(b 1)σ(b 2)\chi'(b_1,b_2) = \sigma'(b_1b_2)^{-1}\sigma'(b_1)\sigma'(b_2) and ψ:BAut(i(K))\psi' : B \to Aut(i(K)) is defined by ψ(b)i(k)=σ(b)i(k)σ(b) 1\psi'(b)i(k) = \sigma'(b)i(k)\sigma'(b)^{-1}. Using the natural identifications i:Ki(K)i : K \cong i(K), and i Aut:Aut(i(K))Aut(K)i_{Aut} : Aut(i(K)) \cong Aut(K), we have ψ=i Autψ\psi' = i_{Aut}\circ \psi and χ=iχ\chi' = i \circ \chi. Now

χ=iχ (b 1,e)(b 2,e)(e,χ(e,e) 1k)=(b 1b 2,e)(e,χ(e,e) 1χ(b 1,b 2)k) (b 1b 2,χ(b 1,b 2))(e,χ(e,e) 1k)=(b 1b 2,χ(b 1b 2,e)χ(e,e) 1χ(b 1,b 2)k) χ(b 1b 2,e)ψ(e) 1(χ(b 1,b 2))χ(e,e) 1k=χ(b 1b 2,e)χ(e,e) 1χ(b 1,b 2)k\array{ \chi'=i\circ\chi &\Leftrightarrow&(b_1,e)(b_2,e)(e,\chi(e,e)^{-1}k) =(b_1 b_2,e)(e,\chi(e,e)^{-1}\chi(b_1,b_2)k)\\ &\Leftrightarrow& (b_1b_2,\chi(b_1,b_2))(e,\chi(e,e)^{-1}k) = (b_1 b_2,\chi(b_1 b_2,e)\chi(e,e)^{-1}\chi(b_1,b_2)k)\\ &\Leftrightarrow& \chi(b_1 b_2,e)\psi(e)^{-1}(\chi(b_1,b_2))\chi(e,e)^{-1}k = \chi(b_1 b_2,e)\chi(e,e)^{-1}\chi(b_1,b_2)k }

for all b 1,b 2Bb_1,b_2 \in B for all kKk \in K in all these lines. The last line is true by (7).

Similarly, ψ=i Autψ\psi' = i_{Aut} \circ \psi iff (b,e)(e,χ(e,e) 1k)=(e,χ(e,e) 1ψ(b)k)(b,e)(b,e)(e,\chi(e,e)^{-1}k) = (e,\chi(e,e)^{-1}\psi(b)k)(b,e) for all bb and kk.

Here the LHS computes as (b,k)(b,k) using χ(b,e)=χ(e,e)\chi(b,e) = \chi(e,e), and the RHS is

(e,ψ(b)k)(b,e)=(b,χ(e,b)ψ(b) 1(χ(e,e) 1ψ(b)(k)))=(b,k)(e,\psi(b)k)(b,e) = (b, \chi(e,b)\psi(b)^{-1}(\chi(e,e)^{-1}\psi(b)(k))) = (b, k)

by (10).

Proposition

The following are equivalent

  • (i) extension (1) is split

  • (ii) for extension (1) there is a subgroup B 1GB_1 \subset G such that B 1i(K)=1B_1 \cap i(K) = 1 and B 1i(K)=GB_1i(K) = G (GG is an internal semidirect product of KK and B 1B_1).

  • (iii) extension (1) is isomorphic to an external semidirect product of KK and BB.

Proof

(i) \Rightarrow (ii) If the extension is split then there is a homomorphism σ:BG\sigma : B \rightarrow G such that pσ=id Bp \circ \sigma = id_B. Let B 1=σ(B)B_1 = \sigma(B). By exactness of (1)), all elements in i(K)i(K) map pp sends to 1, and by pσ=id Bp \circ \sigma = id_B map p B 1p|_{B_1} is injection, therefore the only element in i(K)i(K) which belongs to B 1B_1 is 1.

B 1i(K)=GB_1i(K) = G is also obvious: e.g. for given gG,g \in G, p(g)=pσp(g)p(g) = p\sigma p(g), so that p((σp(g)) 1g)=1p((\sigma p (g))^{-1}g) = 1 what means (σp(g)) 1gKer(p)(\sigma p (g))^{-1}g \in {Ker}(p) so that g=(σp(g))i(k)g = (\sigma p (g))i(k) for some kKk \in K by exactness.

(ii) \Rightarrow (iii) Our previous elaborate discussion of cocycle cross products makes it obvious: choosing a section σ\sigma which is a homomorphism gives χ(a,b)=1\chi(a,b) = 1, and we can construct equivalent external semidirect product as a cocycle cross product with trivial χ\chi.

(iii) \Rightarrow (i) Equivalence of extensions preserves the property of the corresponding short exact sequence to be split. Every external semidirect product is as a set K×BK\times B and the product is given by formula (6) without a cocycle. The map σ:BG\sigma : B \rightarrow G, Bb(1 K,b)K×BB \ni b \mapsto (1_K,b) \in K \times B, splits the sequence.

Definition

An extension (1) is Abelian iff KK is Abelian. An Abelian extension (1) is central iff it is isomorphic to a cocycle cross product extension with all the automorphisms ψ(b),bB\psi(b), b \in B trivial. We say that the extension (1) is Abelian iff GG is Abelian.

Remarks. (i) Note that (8) implies that ψ\psi is a homomorphism if KK in the case of Abelian extensions (for any choice of set-theoretic section σ\sigma.

(ii) If GG is Abelian then (1) is central, but not every central extension is corresponding to an Abelian GG. Abelian extensions in terms of the above definition trivially (strictly!) include both central extensions and extensions with GG central. By abuse of language one sometimes says for GG to be an extension of KK what leads to strange expression that not every Abelian extension (as extension – in terms of the definition above) is Abelian (as a group).

Comparing different extensions; 2-coboundaries

Let us now investigate when two extensions G 1G_1 and G 2G_2 of BB by KK, given by ψ,χ\psi,\chi and ψ,χ\psi',\chi' respectively, are equivalent, cf. diagram (2).

We know that ϵ K:i(k)ϵi(k)\epsilon|_K : i(k) \stackrel{\epsilon}\mapsto i'(k), for all kKk \in K. The formula for ii in \luse{crossform} says that whenever we represent an extension as a cocycle extension we have i(k)=(e,χ(e,e) 1k).i(k) = (e,\chi(e,e)^{-1}k). Thus ϵ(e,χ(e,e) 1k)=(e,χ(e,e) 1k)\epsilon(e,\chi(e,e)^{-1}k) = (e,\chi'(e,e)^{-1}k), for all kK.k \in K. Also recall (or recalculate) that every element (a,k)(a,k) in GG can be factorized as (a,e)(e,χ(e,e) 1k)(a,e)(e,\chi(e,e)^{-1}k). By the definition ϵ\epsilon is a homomorphism of groups, so ϵ(a,k)=ϵ(a,e)ϵ(e,χ(e,e) 1k)\epsilon(a,k) = \epsilon(a,e)\epsilon(e,\chi(e,e)^{-1}k). Also the cosets are preserved, so ϵ(a,e)=(a,λ(a))\epsilon(a,e) = (a,\lambda(a)) where λ:BK\lambda : B \rightarrow K is some set-theoretic map. Thus

ϵ(a,k) = (a,λ(a))(e,χ(e,e) 1k) = (a,χ(a,e)ψ(e) 1(λ(a))χ(e,e) 1k) = (a,λ(a)k).\array{ \epsilon(a,k) & = & (a,\lambda(a))(e,\chi'(e,e)^{-1}k) \\ & = & (a, \chi'(a,e)\psi'(e)^{-1}(\lambda(a))\chi'(e,e)^{-1}k) \\ & = & (a, \lambda(a)k). }

Now multiply more general elements in GG:

ϵ((b 1,k 1)(b 2,k 2))=(b 1,λ(b 1)k 1)(b 2,λ(b 2)k 2) =(b 1b 2,χ(b 1,b 2)ψ(b 2) 1(λ(b 1)k 1)λ(b 2)k 2) \array{ \epsilon((b_1,k_1)(b_2,k_2)) = (b_1,\lambda(b_1)k_1)(b_2,\lambda(b_2)k_2) \\ = (b_1b_2,\chi'(b_1,b_2)\psi'(b_2)^{-1}(\lambda(b_1)k_1)\lambda(b_2)k_2) }

what should be the same as

(11)ϵ((b 1b 2,χ(b 1,b 2)ψ(b 2) 1(k 1)k 2))=(b 1b 2,λ(b 1b 2)χ(b 1b 2)ψ(b 2) 1(k 1)k 2) \epsilon((b_1b_2,\chi(b_1,b_2)\psi(b_2)^{-1}(k_1)k_2)) = (b_1b_2,\lambda(b_1b_2)\chi(b_1b_2)\psi(b_2)^{-1}(k_1)k_2)

In a special case, when k 1=e Kk_1 = e_K we have therefore

(12)χ(b 1,b 2)=λ(b 1b 2) 1χ(b 1,b 2)ψ(b 2) 1(λ(b 1))λ(b 2) \chi(b_1,b_2) = \lambda(b_1b_2)^{-1}\chi'(b_1,b_2) \psi'(b_2)^{-1}(\lambda(b_1))\lambda(b_2)

In order to obtain a relation between ψ(b)(k)\psi'(b)(k) and ψ(b)(k)\psi(b)(k) note that

(13)ϵ((e,χ(e,e) 1k)(b,e))=(e,χ(e,e) 1k)(b,λ(b)). \epsilon ((e,\chi(e,e)^{-1}k)(b,e)) = (e, \chi'(e,e)^{-1}k)(b,\lambda(b)).

That is equivalent to any in the following chain of formulas:

ϵ(b,χ(e,b)ψ(b) 1(χ(e,e) 1k)) = (b,ψ(b) 1(χ(e,e) 1k)λ(b)) λ(b)χ(e,b)ψ(b) 1(χ(e,e) 1k)) = χ(e,b)ψ(b) 1(χ(e,e) 1k)λ(b)\array{ \epsilon (b,\chi(e,b)\psi(b)^{-1}(\chi(e,e)^{-1}k)) &=& (b, \psi'(b)^{-1}(\chi'(e,e)^{-1}k)\lambda(b)) \\ \Leftrightarrow \lambda(b)\chi(e,b)\psi(b)^{-1}(\chi(e,e)^{-1}k)) &=& \chi'(e,b)\psi'(b)^{-1}(\chi'(e,e)^{-1}k)\lambda(b) }

Then by (10), it follows that

λ(b)χ(e,b)χ(e,b) 1ψ(b) 1(k) = χ(e,b)χ(e,b) 1ψ(b) 1(k)λ(b) λ(b)ψ(b) 1(k)) = (ψ(b) 1(k))λ(b) (Ad K(λ(b))ψ(b) 1)(k) = ψ(b) 1(k)\array{ \lambda(b)\chi(e,b)\chi(e,b)^{-1}\psi(b)^{-1}(k) &=& \chi'(e,b)\chi'(e,b)^{-1}\psi'(b)^{-1}(k)\lambda(b) \\ \Leftrightarrow \lambda(b)\psi(b)^{-1}(k)) &=& (\psi'(b)^{-1}(k))\lambda(b) \\ \Leftrightarrow (Ad_K(\lambda(b)) \circ\psi(b)^{-1})(k) &=& \psi'(b)^{-1}(k) }

Now invert the maps in Aut(K)Aut(K) to obtain

(14)ψ(b)=ψ(b)Ad K(λ(b) 1) \psi'(b) = \psi(b)Ad_K(\lambda(b)^{-1})

Thus we obtain

Theorem

Two extensions of a group BB by group KK with corresponding systems (ψ,χ)(\psi,\chi) and (ψ,χ)(\psi',\chi') are equivalent iff there is a [[homomorphism}} λ:BK\lambda: B \rightarrow K such that the relations (12) and (14) are valid.

Proof

If function λ\lambda takes values in the center of BB then (14) implies that ψ=ψ:BAut(K)\psi' = \psi : B \rightarrow Aut(K) and conversely.

If instead of functions ψ\psi and ψ\psi' we consider the respective maps into the group of external automorphisms (cosets of automorphisms with respect to the group of internal homomorphisms) [ψ],[ψ]:~BAut(K)/Int(K)[\psi], [\psi']:~B \rightarrow Aut(K)/Int(K), then the equivalent extensions define the same maps. By (8) these maps are actually homomorphisms (unlike e.g.ψ\psi). For a given ψ\psi if there is χ\chi so that (ψ,χ)(\psi,\chi) does define an extension of BB by KK we say that the extension is associated to (the homomorphism) [ψ][\psi]. That does not mean that any given homomorphism in hom Group(B,Aut(K)/Int(K))hom_{Group}(B,Aut(K)/Int(K)) is associated to any extension, nor it means that if a homomorphism is associated to some extension, that every its representative in hom Set(B,Aut(K))hom_{Set}(B,Aut(K)) is a part of a pair (ψ,χ)(\psi,\chi) defining an extension. To see that situation in more detail we start with a given automorphism, which we call θ\theta , and choose an element ψ(a)θ(a)\psi(a)\in\theta(a), the representative of a coset in Aut(K)/Int(K)Aut(K)/Int(K); that choice should be specified for all aBa \in B. Note that for any ρAut(K),aK\rho \in Aut(K), a \in K we have, by direct inspection, ρAd K(a)ρ 1=Ad K(ρ(a))\rho Ad_K(a)\rho^{-1} = Ad_K(\rho(a)). Thus there is a well-defined function

Ad Kh:B×BInt(K),(Ad Kh)(a,b):=ψ(ab) 1ψ(a)ψ(b) Ad_K \circ h : B \times B \rightarrow Int(K), \,\,\, (Ad_K\circ h)(a,b) := \psi(a b)^{-1}\psi(a)\psi(b)
  • indeed
(15)ψ(a)Ad K(r 1)ψ(b)Ad K(r 2)=ψ(a)ψ(b)Ad K(ψ(b) 1(r 1)r 2)\psi(a)Ad_K(r_1)\psi(b)Ad_K(r_2) = \psi(a)\psi(b)Ad_K(\psi(b)^{-1}(r_1)r_2)

so choosing ψ(ab)[ψ(ab)]\psi(a b) \in [\psi(a b)] is the same as choosing it in [ψ(a)][ψ(b)][\psi(a)][\psi(b)] and guarantees that ψ(ab) 1ψ(a)ψ(b)\psi(a b)^{-1}\psi(a)\psi(b) is in Int(K)Int(K). Let us choose some hh so that Ad KhAd_K \circ h is interpretable as a genuine composition.

(ψ(a)ψ(b))ψ(c) = ψ(ab)Ad K(h(a,b))ψ(c) = ψ(ab)ψ(c)ψ(c) 1Ad K(h(a,b))ψ(c) = ψ(abc)Ad K(h(ab,c))Ad K(ψ(c) 1h(a,b)) = ψ(abc)Ad K(h(ab,c)ψ(c) 1h(a,b))\array{ (\psi(a)\psi(b))\psi(c) & = & \psi(a b)Ad_K(h(a,b))\psi(c) \\ & = & \psi(a b)\psi(c)\psi(c)^{-1}Ad_K(h(a,b))\psi(c) \\ & = & \psi(a b c)Ad_K(h(a b,c))Ad_K(\psi(c)^{-1}h(a,b)) \\ & = & \psi(a b c)Ad_K(h(a b,c)\psi(c)^{-1}h(a,b)) }

what is by associativity the same as

ψ(a)(ψ(b)ψ(c)) = ψ(a)ψ(bc)Ad K(h(b,c)) = ψ(abc)Ad K(h(a,bc))Ad K(h(b,c)) = ψ(abc)Ad K(h(a,bc)h(b,c)).\array{ \psi(a)(\psi(b)\psi(c)) & = & \psi(a)\psi(b c)Ad_K(h(b,c)) \\ & = & \psi(a b c)Ad_K(h(a,b c))Ad_K(h(b,c)) \\ & = & \psi(a b c)Ad_K(h(a,b c)h(b,c)). }

Thus Ad K(h(ab,c)ψ(c) 1h(a,b))=Ad K(h(a,bc)h(b,c)).Ad_K(h(a b,c)\psi(c)^{-1}h(a,b)) = Ad_K(h(a,b c)h(b,c)). Two elements of KK generate the same automorphism iff they differ by a central element. Thus

(16)h(ab,c)ψ(c) 1h(a,b)=h(a,bc)h(b,c)z(a,b,c) h(a b,c)\psi(c)^{-1}h(a,b) = h(a,b c)h(b,c)z(a,b,c)

for a unique central element z(a,b,c)Z(K).z(a,b,c) \in Z(K). The correspondence z:(a,b,c)z(a,b,c)z : (a,b,c) \mapsto z(a,b,c) maps B×B×BB \times B \times B into Z(K)Z(K).

Proposition

zz is an (Abelian) 3-cocycle with values in /Z(K) ψ 1_/Z(K)_{\psi^{-1}} (Z(K)Z(K) understood as trivial-ψ 1\psi^{-1} BB-bimodule):

(17)z(b,c,d)z(a,bc,d)ψ(d) 1z(a,b,c)=z(a,b,cd)z(ab,c,d) z(b,c,d)z(a,b c,d)\psi(d)^{-1}z(a,b,c) = z(a,b,c d)z(a b,c,d)
Proof

To see this we calcuate

h(abc,d)[ψ(d) 1h(ab,c)ψ(c) 1h(a,b)] =h(abc,d)[ψ(d) 1h(a,bc)h(b,c)z(a,b,c)] =h(a,bcd)h(bc,d)z(a,bc,d)[ψ(d) 1h(b,c)z(a,b,c)] =h(a,bcd)h(b,cd)h(c,d)z(b,c,d)z(a,bc,d)ψ(d) 1z(a,b,c)\array{ h(a b c,d)[\psi(d)^{-1}h(a b,c)\psi(c)^{-1}h(a,b)] & = h(a b c,d)[\psi(d)^{-1}h(a,b c)h(b,c)z(a,b,c)] \\ & = h(a,b c d)h(b c,d)z(a,b c,d)[\psi(d)^{-1}h(b,c)z(a,b,c)] \\ & = h(a,b c d)h(b,c d)h(c,d)z(b,c,d)z(a,b c,d)\psi(d)^{-1}z(a,b,c) }

Compare

h(abc,d)[ψ(d) 1h(ab,c)ψ(c) 1h(a,b)] =h(abc,d)[ψ(d) 1h(a,bc)]ψ(d) 1ψ(c) 1h(a,b) =h(abc,d)[ψ(d) 1h(ab,c)]h(c,d) 1[ψ(cd) 1h(a,b)]h(c,d) =h(abc,d)h(ab,cd)z(ab,c,d)[ψ(cd) 1h(a,b)]h(c,d) =h(a,bcd)h(b,cd)h(c,d)z(a,b,cd)z(ab,c,d)\array{ h(a b c,d)[\psi(d)^{-1}h(a b,c)\psi(c)^{-1}h(a,b)] & = h(a b c,d)[\psi(d)^{-1}h(a,b c)]\psi(d)^{-1}\psi(c)^{-1}h(a,b) \\ & = h(a b c,d)[\psi(d)^{-1}h(a b,c)]h(c,d)^{-1}[\psi(c d)^{-1}h(a,b)]h(c,d) \\ & = h(a b c,d)h(a b,c d)z(a b,c,d)[\psi(c d)^{-1}h(a,b)]h(c,d) \\ & = h(a,b c d)h(b,c d)h(c,d)z(a,b,c d)z(a b,c,d) }
Proposition

(i) If we choose a different hh such that

Ad K(h(a,b))=ψ(ab) 1ψ(a)ψ(b),Ad_K(h(a,b)) = \psi(a b)^{-1}\psi(a)\psi(b),

then zz will change only up to a 3-coboundary df,d f, i.e. there is a function f:B×BZ(K)f : B \times B \rightarrow Z(K), such that z=(df)zz' = (d f)z where

(18)(df)(a,b,c)=f 1(b,c)f 1(a,bc)f(ab,c)ψ(c) 1f(a,b),foralla,b,cB. (d f)(a,b,c) = f^{-1}(b,c)f^{-1}(a,b c)f(a b,c)\psi(c)^{-1}f(a,b),\,\,\,\,\, for all a,b,c \in B.

(ii) Conversely, if zz is a 3-cocycle obtained from ψ\psi as above and dfd f is a 3-coboundary, then there is a hh' determining the same inner automoprhism of KK such that the corresponding 3-cocycle zz' is equal to (df)z(df)z.

(iii) Let ψ,ψ:BAut(K)\psi, \psi' : B \rightarrow Aut(K) be two set-theoretic sections so that [ψ]=[ψ]=θ:BAut(K)/Int(K)[\psi] = [\psi'] = \theta : B \rightarrow Aut(K)/Int(K), then (for arbitrary choice of hh, hh') the cocycles zz and zz' obtained as above differ only up to a 3-coboundary. \|

Proof

(i) Choose two different h,h:B×BKh',h: B \times B \rightarrow K such that Ad K(h)=Ad K(h)Ad_K(h') = Ad_K(h). Then h(a,b)=h(a,b)f(a,b)h'(a,b) = h(a,b)f(a,b) where f:B×BZ(K)f : B \times B \rightarrow Z(K) is some function with values in center of KK. A direct comparison of (16) written for h,zh,z and h,zh',z' respectively proves the assertion.

(ii) Trivial: Any f:B×BZ(K)f : B \times B \rightarrow Z(K) such that h=hfh' = hf will not change the inner automorphism. Thus any central 3-coboundary dfdf can be obtained by changing a choice of hh.

(iii) [ψ]=[ψ][\psi'] = [\psi] implies that exists k:BK,ψ(a)=ψ(a)Ad K(k(a)).k : B \rightarrow K, \psi'(a) = \psi(a)Ad_K(k(a)). Then

ψ(ab)Ad K(h(a,b)) = ψ(a)ψ(b)=ψ(a)Ad K(k(a))ψ(b)Ad K(k(b)) = ψ(a)ψ(b)Ad K([ψ(b) 1k(a)]k(b)) = ψ(ab)Ad K(h(a,b)[ψ(b) 1k(a)]k(b)) = ψ(ab)Ad K(k(ab) 1h(a,b)[ψ(b) 1k(a)]k(b)).\array{ \psi'(a b)Ad_K(h'(a,b)) & = & \psi'(a)\psi'(b) = \psi(a)Ad_K(k(a))\psi(b)Ad_K(k(b))\\ & = & \psi(a)\psi(b)Ad_K([\psi(b)^{-1}k(a)]k(b)) \\ & = & \psi(a b)Ad_K(h(a,b)[\psi(b)^{-1}k(a)]k(b)) \\ & = & \psi'(a b)Ad_K(k(a b)^{-1}h(a,b)[\psi(b)^{-1}k(a)]k(b)). }

Thus h(a,b)=k(ab) 1h(a,b)[ψ(b) 1k(a)]k(b), h'(a,b) = k(a b)^{-1}h(a,b)[\psi(b)^{-1}k(a)]k(b), for appropriate choice of hh' - what can change zz' up to coboundary - using the freedom from (i). If we want formula involving ψ\psi' instead than we use ψ(a)=ψ(a)Ad K(k(a))\psi'(a) = \psi(a)Ad_K(k(a)) to obtain k(ab)h(a,b)=h(a,b)k(b)[ψ(b) 1k(a)]k(a b)h'(a,b) = h(a,b)k(b)[\psi'(b)^{-1}k(a)]. Using that and previous identities,

k(abc)h(ab,c)ψ(c) 1h(a,b) = h(ab,c)k(c)[ψ(c) 1k(ab)]ψ(c) 1h(a,b) = h(ab,c)k(c)ψ(c) 1k(ab)h(a,b) = h(ab,c)k(c)ψ(c) 1h(a,b)k(b)[ψ(b) 1k(a)] = h(ab,c)[ψ(c) 1h(a,b)]k(c)ψ(c) 1k(b)[ψ(b) 1k(a)] = h(a,bc)h(b,c)z(a,b,c)k(c)[ψ(c) 1k(b)][ψ(c) 1ψ(b) 1k(a)] = h(a,bc)h(b,c)k(c)[ψ(c) 1k(b)][ψ(c) 1ψ(b) 1k(a)]z(a,b,c) = h(a,bc)k(bc)h(b,c)[ψ(c) 1ψ(b) 1k(a)]z(a,b,c) = h(a,bc)k(bc)h(b,c)h(b,c) 1[ψ(bc) 1k(a)]h(b,c)z(a,b,c) = h(a,bc)k(bc)[ψ(bc) 1k(a)]h(b,c)z(a,b,c) = k(abc)h(a,bc)h(b,c)z(a,b,c)\array{ k(a b c)h'(a b,c)\psi'(c)^{-1}h'(a,b) &=& h(a b,c)k(c)[\psi'(c)^{-1}k(a b)]\psi'(c)^{-1}h'(a,b) \\ &=& h(a b,c)k(c)\psi'(c)^{-1}k(a b)h'(a,b) \\ &=& h(a b,c)k(c)\psi'(c)^{-1}h(a,b)k(b)[\psi'(b)^{-1}k(a)] \\ &=& h(a b,c)[\psi(c)^{-1}h(a,b)]k(c)\psi'(c)^{-1}k(b)[\psi'(b)^{-1}k(a)] \\ &=& h(a,b c)h(b,c)z(a,b,c)k(c) [\psi'(c)^{-1}k(b)][\psi'(c)^{-1}\psi'(b)^{-1}k(a)] \\ &=& h(a,b c)h(b,c)k(c)[\psi'(c)^{-1}k(b)] [\psi'(c)^{-1}\psi'(b)^{-1}k(a)]z(a,b,c) \\ &=& h(a,b c)k(b c)h'(b,c)[\psi'(c)^{-1}\psi'(b)^{-1}k(a)]z(a,b,c) \\ &=& h(a,b c)k(b c)h'(b,c)h'(b,c)^{-1}[\psi'(b c)^{-1}k(a)]h'(b,c)z(a,b,c)\\ &=& h(a,b c)k(b c)[\psi'(b c)^{-1}k(a)]h'(b,c)z(a,b,c)\\ &=& k(a b c)h'(a,b c)h'(b,c)z(a,b,c) }

for all a,b,cBa,b,c \in B. Thus h(ab,c)ψ(c) 1h(a,b)=h(a,bc)h(b,c)z(a,b,c)h'(a b,c)\psi'(c)^{-1}h'(a,b) = h'(a,b c)h'(b,c)z(a,b,c) i.e. our choice of hh' insured no change in zz. Of course that means that in arbitrary choice of hh' we do not miss more than a coboundary by (i).

Corollary

A given homomorphism θ:B×BAut(K)/Int(K)\theta : B \times B \rightarrow Aut(K)/Int(K) is associated to some extension of BB by KK iff zz is a 3-coboundary.

Proof

Indeed, if θ\theta is associated to an extension, then we know that there is an isomorphism of the extension with a cross product given by some cocycle χ\chi and some automorphism ψ\psi such that [ψ]=θ[\psi] = \theta. But using the identification, χ=h\chi = h for that particular choice of ψ\psi, so that z=1z = 1. By the proposition, every other zz obtained from θ\theta is in the same cohomology class, thus every such zz is a coboundary. Conversely, if zz is a coboundary, then by the proposition, we can change it to z=1z = 1, and then we have all the conditions for a cross product extension satisfied.

Formulation in homotopy theory

One may regard the above from the nPOV as a special case of the way cocycles in the general notion of cohomology classify their homotopy fibers. More on this is at

Examples

By the above classification theorems, all the examples at group cohomology equivalently induce examples for group extensions. And indeed by definition every short exact sequence defines an extension.

But examples of fundamental importance include for instance

References

General

  • Samuel Eilenberg, Saunders MacLane, Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel. Ann. of Math. (2) 48, (1947). 326–341 jstor

  • Saunders MacLane, Cohomology theory in abstract groups. III. Operator homomorphisms of kernels. Ann. of Math. (2) 50, (1949). 736–761.

  • Lawrence Breen, Théorie de Schreier supérieure, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 465–514 numdam.

Textbooks include

  • A. G. Kurosh, Theory of groups

  • Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, 87, Springer-Verlag, New York-Berlin, 1982.

Lecture notes and similar include

See also

  • R. Brown, T. Porter, On the Schreier theory of non-abelian extensions: generalisations and computations, Proc. Roy. Irish Acad. Sect. A, 96, (1996), 213 – 227.

  • Manuel Bullejos, Antonio M. Cegarra, A 3-dimensional non-abelian cohomology of groups with applications to homotopy classification of continuous maps Canad. J. Math., vol. 43, (2), 1991, p. 1-32.

  • Antonio M. Cegarra, Antonio R. Garzón, A long exact sequence in non-abelian cohomology, Proc. Int. Conf. Como 1990., Lec. Notes in Math. 1488, Springer 1991.

See also references of Dedecker listed here.

Applications

A bit of discussion of some occurences of central extensions of groups in physics is in

  • G. Tuynman and W. Wiegerinck, Central extensions of physics (pdf)

(In fact there are many more than mentioned in that introduction.)

Revised on November 7, 2013 03:15:52 by Urs Schreiber (77.251.114.72)