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group cohomology, nonabelian group cohomology, Lie group cohomology
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A group extension of a group $G$ by a group $A$ is third group $\hat G$ that sits in a short exact sequence, that can usefully be thought of as a fiber sequence, $A \to \hat G \to G$.
Two consecutive homomorphisms of groups
are a short exact sequence if
$i$ is monomorphism,
$p$ an epimorphism
the image of $i$ is all of the kernel of $p$: $ker(p)\simeq im(i)$.
We say that such a short exact sequence exhibits $\hat G$ as an extension of $G$ by $A$.
If $A \hookrightarrow \hat G$ factors through the center of $\hat G$ we say that this is a central extension.
Sometimes in the literature one sees $\hat G$ called an extension “of $A$ by $G$”. 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. (On the other hand, our terminology conflicts with the usual meaning of “extension” in algebra. For example, in Galois theory if $k$ is a field, then an extension of $k$ contains $k$ as a subfield.)
Under the looping and delooping-equivalence, this is equivalently reformulated as follows. For $G \in$ Grp a group, write $\mathbf{B}G \in$ Grpd for its delooping groupoid.
A sequence $A \to \hat G \to G$ is a short exact sequence of groups precisely if the delooping
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.
A homomorphism of extensions $f : \hat G_1 \to \hat G_2$ of a given $G$ by a given $A$ is a group homomorphism of this form which fits into a commuting diagram
A morphism of extensions as in def. is necessarily an isomorphism.
By the short five lemma.
We discuss properties of group extensions in stages,
For $A \hookrightarrow \hat G \to G$ a group extension, the inclusion $A \hookrightarrow \hat G$ is a normal subgroup inclusion.
We need to check that for all $a \in A \hookrightarrow G$ and $g \in G$ the result of the adjoint action $g a g^{-1}$ formed in $\hat G$ is again in $A \stackrel{i}{\hookrightarrow} \hat G$.
Since $p : \hat G \to G$ is a group homomorphism we have that
and hence $g a g^{-1}$ is in the kernel of $p$. By the defining exactness property therefore it is in the image of $i$.
For $A \stackrel{i}{\to} \hat G \stackrel{p}{\to} G$ a group extension, we have that $p : \hat G \to G$ is an $A$-torsor over $G$ where the action of $A$ on $\hat G$ is defined by
That $\rho$ is indeed an action over $B$ in that
follows from the fact that $p$ is a group homomorphism and that $A$ is in its kernel.
That $A$ is actually equal to the kernel gives the principality condition
For $A$ an abelian group we may understand the $A$-torsor/$A$-principal bundle $\hat G$ as the delooping of the $\mathbf{B}A$-principal 2-bundle $\mathbf{B} \hat G \to \mathbf{BG}$ that is classified by (is the homotopy fiber of) the 2-cocycle in group cohomology $c : \mathbf{B}G \to \mathbf{B}^2 A$ that classifies the extension.
All this is then summarized by the statement that
is a fiber sequence in ∞Grpd (or in ?LieGrpd? if we have Lie group extensions, etc).
Here we may think of $\mathbf{B}\hat G$ as being the $\mathbf{B}A$-principal 2-bundle over $\mathbf{B}G$ classified by $c$. See the examples discussed at bundle gerbe.
A group extension $A \to \hat G \stackrel{p}{\to} G$ is called split if there is a section homomorphism $\sigma \colon G \to \hat G$, hence a group homomorphism such that $p \circ \sigma = id_G$.
It is important here that $\sigma$ 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 $p$ is an epimorphism by definition.
Split extensions $\hat G$ of $G$ by $A$, def. , are equivalently semidirect product groups
for some action $\rho \colon G \times A \to A$ of $G$ on $A$.
This means that the underlying set is $U(A \rtimes_\rho G) = U(A) \times U(G)$ and the group operation in $A \rtimes_\rho G$ is
The inclusion of $A$ is by
and the projection to $G$ is by
Given a split extension $A \stackrel{i}{\to} \hat G \stackrel{p}{\to} G$ with splitting $\sigma \colon G \to \hat G$, define an action of $G$ on $A$ by the restriction of the adjoint action $\rho_{ad}$ of $\hat G$ on itself to $A$:
Then (…)
A split extension $A \to \hat G \to G$ is a central extension precisely if the action $\rho$ induced from it as in prop. is trivial.
For it to be a central extension the inclusion $A \to A \rtimes_\rho G$ has to land in the center of $A \rtimes_\rho G$, hence all elements $a \in A$ have to commute as elements $(a,e) \in A \rtimes_\rho G$ with all elements of $A \rtimes_\rho G$. But consider elements of the form $(e,g) \in A \rtimes_\rho G$ for all $g \in G$. Then
but
For these to be equal for all $a \in A$, $\rho(g)$ has to be the identity. Since this is to be true for all $g \in G$, the action has to be trivial.
This means in particular that split central extensions are product groups $A \to G$. If all groups involved are abelian groups, then these are equivalently the direct sums $A \oplus G$ of abelian groups. In this way the notion of split group extension reduces to that of split short exact sequences of abelian groups.
If we have a split extension the different splittings are given by derivations, but with possibly non-abelian values. In fact if we have $s: G\to A\rtimes G$ is a section then $s(g) = (a(g),g)$, and the multiplication in $A\rtimes G$ implies that $a: G\to A$ is a derivation. These are considered as the (possibly non-abelian) 1-cocycles of $G$ with (twisted) coefficients in $A$, as considered in, for instance, Serre’s notes on Galois cohomology.
We discuss properties of central group extensions, those where $A \hookrightarrow \hat G$ factors through the center of $\hat G$. 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.
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 $G$ a group and $A$ an abelian group, write
for the degree-2 group cohomology of $G$ with coefficients in $A$, and write
for the group of central extensions of $G$ by $A$.
We prove this below as prop. . Here we first introduce stepwise the ingredients that go into the proof.
(central extension associated to group 2-cocycle)
For $[c] \in H^2_{Grp}(G,A)$ a group cohomology class represented by a cocycle $c \colon G \times G \to A$, define a group
as follows. The underlying set is the cartesian product $U(G) \times U(A)$ of the underlying sets of $G$ and $A$. The group operation on this is given by
This defines indeed a group: the cocycle condition on $c$ gives precisely the associativity of the product on $G \times_c A$. Moreover, the construction extends to a homomorphism of groups
Forming the product of three elements of $G \times_c A$ bracketed to the left is, according to def. ,
Bracketing the same three elements to the right yields
The difference between the two expressions is read off to be precisely
where $d c$ denotes the group cohomology differential of $c$. Hence this vanishes precisely if $c$ is a group 2-cocycle, hence we have an associative product.
To see that it has inverses, notice that for all $(g,a)$ we have
and hence inverses are given by $(g,a)^{-1} = (g^{-1}, -a - c(g,g^{-1}))$. Hence $G \times_c A$ is indeed a group.
By the discussion at group cohomology – degree-2 we may assume without restriction that $c$ is a normalized cocycle, hence that $c(e,-) = c(-,e) = 0$. Using this we find that the inclusion
given by $a \mapsto (e,a)$ is a group homomorphism. Moreover, the projection on the underlying sets evidently yields a group homomorphism $p \colon G \times_c A \to G$ given by $(g,a) \mapsto g$. The kernel of this is $A$, and hence
is indeed a group extension. It is a central extension again using the assumption that $c$ is normalized $c(g,e) = c(e,g) = 0$:
Finally to see that the construction is independent of the choice of coycle $c$ representing $[c]$, let $\tilde c$ be another representative which differs by a coboundary $h \colon G \to A$ with
We claim that then we have a homomorphism of central extensions (hence an isomorphism) of the form
To see this we check for all elements that
Hence the construction of $G \times_c A$ indeed defines a function $H^2_{Grp}(G,A) \to CentrExt(G,A)$.
Assume the axiom of choice in the ambient foundations.
(2-cocycle extracted from central extension)
Given a central extension $A \to \hat G \to G$ define a group 2-cocycle $c : G \times G \to A$ as follows.
Choose a section $\sigma : U(G) \to U(\hat G)$ of the underlying sets (which exists by the axiom of choice and the fact that $p : \hat G \to G$ is by definition an epimorphism). Then define $c$ by
where on the right we are using that by the section-property of $\sigma$ and the group homomorphism property of $p$
and hence by the exactness of the extension the argument is in $A \hookrightarrow \hat G$.
Below in remark is a discussion of how this construction arises from a more systematic discussion in homotopy theory.
The construction of prop. indeed yields a 2-cocycle in group cohomology. It extends to a morphism
The cocycle condition to be checked is that
for all $g_0, g_1, g_2 \in G$. Writing this out with def. yields
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 $H^2_{grp}(G,A)$ we need to check that for $\tilde \sigma : G \to \hat G$ a different choice of section, the corresponding cocycles differ by a group coboundary $\tilde c - c = d h$. Clearly this is obtained by setting
where we use that the right hand side is in $A \hookrightarrow \hat G$ since because both $\sigma$ and $\tilde \sigma$ are sections of $p$, the image of the right hand under $p$ is the neutral element in $G$.
The two morphisms of def. and def. exhibit the equivalence
Let $[c] \in H^2_{Grp}(G,A)$. Then by construction of $\hat G \coloneqq G \times_c A$ there is a canonical section of the underlying function of sets $U(G \times_c A) \to U(G)$ given by $(id_{U(G)}, 0) U(G) \to U(G) \times U(A)$. The cocycle induced by this section sends
which is $c(g_1, g_2) \in A \hookrightarrow G \times_c A$, and hence this recovers the 2-cocycle that we started with.
This shows that $Extr \circ Rec = id$ and in particular that $Rec$ is a surjection. It is readily seen that the kernel of $Rec$ is trivial, and so it is an equivalence.
We discuss the classification of central group extensions by degree-2 group cohomology in the more abstract context of homotopy theory (via the translation discussed at looping and delooping), complementing the above component-wise discussion.
Let
be a central group extension, def. , hence with $A$ an abelian group included in the center of $G$. Then $A$ is in particular a normal subgroup and hence the homorphism
may be regarded as a crossed module of groups. This is equivalently a strict 2-group structure on the groupoid whose objects are $\hat G$ and whose morphisms are labeled in $A$
Write
for the delooping of this 2-group to a one-object 2-groupoid.
The ∞-nerve (or Duskin nerve) $N \mathbf{B}(A \to \hat G) \in$ sSet of this is a (3-coskeletal) Kan complex that realizes this as a 2-truncated ∞-groupoid.
The obvious strict 2-functor
is an equivalence of 2-groupoids.
One way to see this is to notice that this is a k-surjective functor for all $k \in \{0,1,2,3\}$, hence a weak equivalence in the folk model structure on $\omega$-groupoids. Equivalently, under the nerve the morphism of simplicial sets
is an acyclic Kan fibration, hence a weak equivalence in the standard model structure on simplicial sets.
The extension $A \to \hat G \to G$ sits in a long homotopy fiber sequence in the (∞,1)-category ∞Grpd of the form
which in Kan complexes/simplicial sets is presented by the zigzag of n-functors between strict ∞-groupoid (sequence of 2-anafunctors) of the form
In particular, the induced connecting homomorphism
is the group cohomology cocycle that classifies the delooped extension as a $\mathbf{B}A$-principal 2-bundle.
One sees directly that the morphisms $\mathbf{B}\hat G \to \mathbf{B}G$ and $\mathbf{B}(A \to \hat G ) \to \mathbf{B}^2 A$ as well as their loopings $\hat G \to G$ and $(A \to \hat G) \to G$ 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 $\mathbf{B}(A \to \hat G) \to \mathbf{B}(A \to 1) = \mathbf{B}^2 A$ is manifestly $\mathbf{B} \hat G$, and so on.
is precisely the result of moving set-theoretically through the zigzag
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.
For $A, G \in$ Ab $\hookrightarrow$ Grp even a central extension $\hat G$ of $G$ by $A$ is not necessarily itself an abelian group.
But by prop. above it is so if the group 2-cocycle that classifies the extension is symmetric:
A 2-cocycle $c \colon G \times G \to A$ in group cohomology is symmetric if
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.
Write
for the set (group) of classes of symmetric group 2-cocycles on $G$ with coefficients in $A$.
For $G,A \in Ab \hookrightarrow Grp$, write $Ext(G,A)$ for the subset of equivalence class of abelian group extensions of $G$ by $A$.
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
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
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 $p:G\to B$.
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 $g$ of $G$ defines an inner automorphism $\phi(g)$ of $K$ by $\phi(g)(k) = g k g^{-1}$. The restriction $\phi|_K$ takes (by definition) values in the subgroup $Int(K)$ of inner automorphisms of $K$. In fact $\phi:G\to Inn(G)\subset Aut(K)$ is a homomorphism of groups.
If $g_1$ and $g_2$ are in the same left coset, that is $g_1K = g_2K$, then there is $k \in K$, $g_1 = g_2k$, so that $\forall k' \in K$ we have $\phi(g_1k') = \phi(g_2k k') = \phi(g_2)\phi(k k')$ and therefore $\phi(g_1K) \subset \phi(g_2)Int(K)$. Thus we obtain a well-defined map $\phi_* : G/K \rightarrow Aut(K)/Int(K)$. Choose a set-theoretic section of the projection $p : G \rightarrow B$ and let
Warning. Unlike $\phi$, the map $\psi$ is not a homomorphism of groups.
We attempt to reconstruct $G$ from the knowledge of $\psi$ and $K$. As a set, $G$ can be naturally identified with $B \times K$. Indeed, write each element $g \in G$ as $\sigma(b)k, b \in B, k \in K$ by setting $b = p(g), k = \sigma(p(g))^{-1}g$. Elements $b \in B$ and $k \in K$ in that decomposition are unique, and we get a bijection
whose inverse is the map $g \mapsto (p(g), \sigma(p(g))^{-1}g)$. By means of that bijection, $B \times K$ inherits the group structure from $G$. Let us figure out the multiplication rule on $B \times K.$ If $\sigma(b_1)k_1 = g_1$, and $\sigma(b_2)k_2 = g_2$, then,
Now $p(\sigma(b_1)\sigma(b_2)) = p(b_1b_2)$ so
This formula clearly defines a function $\chi : B \times B \rightarrow K$. In this notation,
and using bijection of $G$ with $B\times K$ this can be expressed in terms of elements in $B\times K$ so that
According to this formula, all the information about the multiplication is encoded in functions $\chi : B \times B \rightarrow Aut(K)$ and $\psi : B \rightarrow Aut(K)$, and we may forget about $\sigma$ at this point. However, not every pair $(\chi,\psi)$ will give some multiplication rule on $B \times K$. Let $a,b,c \in B$, and $e = e_K$ be the unity element in $K$. Then
From the other side, this has to be the same, by associativity, to
where we took into account that expressions like $[\psi^{-1}(b)(e)] = e$, because $\psi(b)$ is an automorphism for each $b \in B$.
Comparing the expressions above we obtain
If the pair $(\chi,\psi)$ is constructed as above, then
where $Ad_K$ is the canonical map $K \rightarrow Int(K)$, $k\mapsto k(-)k^{-1}$.
Thus we obtain the relation
Let $B$ and $K$ be two groups. Let $\chi: B \times B \rightarrow K$ and $\psi : B \rightarrow Aut(K)$ satisfy (4) and (5). Then we call that the family $\{\chi(b_1,b_2)| b_1,b_2 \in B\}$ is a factor system (This term is due Schreier(1924)) or a nonabelian group 2-cocycle with automorphisms, and the family $\{\psi(b) | b \in B \}$ – a system of automorphisms
A 2-cocycle $\chi$ is counital if $\chi(b,e) = \chi(e,b) = e$, for all $b \in B$.
If $K$ is commutative, then $\psi$ is always a homomorphism (cf. (5)). Then $K$ is a right $B$-module through $\psi(-)^{-1}$. That justifies the sometimes used term “(right) cocycle $B$-module” for $(K,\psi,\chi)$. If $\psi$ is trivial ($\psi(b) = Id_K, \forall b \in B$) then the cocycle condition (4) becomes
If formulas (4) and (5) are both satisfied, then the formula (3) for multiplication of pairs defines a group multiplication on $B \times K$. That set, together with multiplication (3) is called the cocycle cross product of $B$ and $K$ with cocycle $\chi$ and action $\psi$. If the cocycle is trivial i.e. $\chi(\cdot,\cdot) = e_K$, we call it the (external) semidirect product.
We have checked above the associativity for pairs of the form $(a,e)$ etc. This was useful to find the cocycle condition correctly. Now the general associativity should be a similar calculation with general elements. Using (4) and (5) it can be done.
where we used (5).
Thus $Ad_K(\chi(a,e)) = \psi(e)$ and therefore it does not depend on $a$.
Then use (4) with $b = c = e$ to get $\psi(e)^{-1}(\chi(a,e)) = \chi(e,e), \forall a \in B$.
Thus $\chi(a,e)^{-1}(\chi(a,e))\chi(a,e) = \chi(e,e)$, that is $\chi(a,e)$ does not depend on $a$.
Now we claim that the unit element is given by $(e, \chi(e,e)^{-1})$. To verify that it is also a right unit we compute
what is equal to $(a,b)$ by just proved statement that $\chi(a,e)$ does not depend on $a$.
Now use (4) with $a = b = e$ to get
Thus we can verify that $(e, \chi(e,e)^{-1})$ is a left unit too by a calculation as follows. Namely
by the definition of the product. Then by (6), this equals to
and, because $\psi(a)^{-1}$ is an antiautomorphism, this is finally equal to $(a,b)$.
Now check that each element $(a,b)$ can be factorized as $(a,e)(e,\chi(e,e)^{-1}b)$. In order to show that $(a,b)$ has an inverse it is then enough to show that both $(a,e)$ and $(e,\chi(e,e)^{-1}b)$ have inverses.
Claim: the inverse of $(a,e)$ is
To this aim, we calculate
because $\psi(a)^{-1}(e) = e$. Furthermore,
because $\psi(a)^{-1}(e) = e$. Next,
what equals $(e,\chi(e,e)^{-1})$.
Indeed, (4) with $a = a, b = a^{-1}, c = a$ reads $\chi(e,a) \psi(a)^{-1}(\chi(a, a^{-1}))$ $=\chi(a,e)\chi(a^{-1},a)$.
Then apply (6) and take inverse of both sides to obtain
Then recall that $\chi(a,e)$ does not depend on $a$ and multiply by $\chi(a^{-1},a)$ from the left.
Claim: the inverse of $(e,\chi(e,e)^{-1}k)$ is $(e,\chi(e,e)k^{-1})$. Here the verification is symmetric ($k$ vs. $k^{-1}$) for the left and for the right inverse and immediate.
Given groups $K$ and $B$ and any maps $\chi$ and $\psi$ satisfying (4) and (5), needed to define a cocycle cross product $B\times_\chi K$ of $K$ and $B$, one defines the map $i : K \rightarrow B \times_\chi K$ by $k \mapsto (e,\chi(e,e)^{-1}k)$. Then $i$ is a monomorphism of groups, $i(K)$ is a normal subroup of the cocycle cross product of $B$ and $K$, and there is a canonical isomorphism $B \cong G/K$. We define the set-theoretic maps $\sigma',\chi'$ and $\psi'$ as follows. $\sigma' : B \rightarrow B \times K$ is defined by $\sigma'(b) = (b, e)$ , for all $b \in B$. Then $\chi' : B \times B \to i(K)$ is defined by $\chi'(b_1,b_2) = \sigma'(b_1b_2)^{-1}\sigma'(b_1)\sigma'(b_2)$ and $\psi' : B \to Aut(i(K))$ is defined by $\psi'(b)i(k) = \sigma'(b)i(k)\sigma'(b)^{-1}$. Using the natural identifications $i : K \cong i(K)$, and $i_{Aut} : Aut(i(K)) \cong Aut(K)$, we have $\psi' = i_{Aut}\circ \psi$ and $\chi' = i \circ \chi$. Now
for all $b_1,b_2 \in B$ for all $k \in K$ in all these lines. The last line is true by (4).
Similarly, $\psi' = i_{Aut} \circ \psi$ iff $(b,e)(e,\chi(e,e)^{-1}k) = (e,\chi(e,e)^{-1}\psi(b)k)(b,e)$ for all $b$ and $k$.
Here the LHS computes as $(b,k)$ using $\chi(b,e) = \chi(e,e)$, and the RHS is
by (6).
The following are equivalent
(i) $\Rightarrow$ (ii) If the extension is split then there is a homomorphism $\sigma : B \rightarrow G$ such that $p \circ \sigma = id_B$. Let $B_1 = \sigma(B)$. By exactness of (1)), all elements in $i(K)$ map $p$ sends to 1, and by $p \circ \sigma = id_B$ map $p|_{B_1}$ is injection, therefore the only element in $i(K)$ which belongs to $B_1$ is 1.
$B_1i(K) = G$ is also obvious: e.g. for given $g \in G,$ $p(g) = p\sigma p(g)$, so that $p((\sigma p (g))^{-1}g) = 1$ what means $(\sigma p (g))^{-1}g \in {Ker}(p)$ so that $g = (\sigma p (g))i(k)$ for some $k \in K$ by exactness.
(ii) $\Rightarrow$ (iii) Our previous elaborate discussion of cocycle cross products makes it obvious: choosing a section $\sigma$ which is a homomorphism gives $\chi(a,b) = 1$, and we can construct equivalent external semidirect product as a cocycle cross product with trivial $\chi$.
(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 $K\times B$ and the product is given by formula (3) without a cocycle. The map $\sigma : B \rightarrow G$, $B \ni b \mapsto (1_K,b) \in K \times B$, splits the sequence.
An extension (1) is Abelian iff $K$ is Abelian. An Abelian extension (1) is central iff it is isomorphic to a cocycle cross product extension with all the automorphisms $\psi(b), b \in B$ trivial. We say that the extension (1) is Abelian iff $G$ is Abelian.
Remarks. (i) Note that (5) implies that $\psi$ is a homomorphism if $K$ in the case of Abelian extensions (for any choice of set-theoretic section $\sigma$.
(ii) If $G$ is Abelian then (1) is central, but not every central extension is corresponding to an Abelian $G$. Abelian extensions in terms of the above definition trivially (strictly!) include both central extensions and extensions with $G$ central. By abuse of language one sometimes says for $G$ to be an extension of $K$ what leads to strange expression that not every Abelian extension (as extension – in terms of the definition above) is Abelian (as a group).
Let us now investigate when two extensions $G_1$ and $G_2$ of $B$ by $K$, given by $\psi,\chi$ and $\psi',\chi'$ respectively, are equivalent, cf. diagram (2).
We know that $\epsilon|_K : i(k) \stackrel{\epsilon}\mapsto i'(k)$, for all $k \in K$. The formula for $i$ in \luse{crossform} says that whenever we represent an extension as a cocycle extension we have $i(k) = (e,\chi(e,e)^{-1}k).$ Thus $\epsilon(e,\chi(e,e)^{-1}k) = (e,\chi'(e,e)^{-1}k)$, for all $k \in K.$ Also recall (or recalculate) that every element $(a,k)$ in $G$ can be factorized as $(a,e)(e,\chi(e,e)^{-1}k)$. By the definition $\epsilon$ is a homomorphism of groups, so $\epsilon(a,k) = \epsilon(a,e)\epsilon(e,\chi(e,e)^{-1}k)$. Also the cosets are preserved, so $\epsilon(a,e) = (a,\lambda(a))$ where $\lambda : B \rightarrow K$ is some set-theoretic map. Thus
Now multiply more general elements in $G$:
what should be the same as
In a special case, when $k_1 = e_K$ we have therefore
In order to obtain a relation between $\psi'(b)(k)$ and $\psi(b)(k)$ note that
That is equivalent to any in the following chain of formulas:
Then by (6), it follows that
Now invert the maps in $Aut(K)$ to obtain
Thus we obtain
Two extensions of a group $B$ by group $K$ with corresponding systems $(\psi,\chi)$ and $(\psi',\chi')$ are equivalent iff there is a homomorphism $\lambda: B \rightarrow K$ such that the relations (7) and (8) are valid.
If function $\lambda$ takes values in the center of $B$ then (8) implies that $\psi' = \psi : B \rightarrow Aut(K)$ and conversely.
If instead of functions $\psi$ and $\psi'$ we consider the respective maps into the group of external automorphisms (cosets of automorphisms with respect to the group of internal homomorphisms) $[\psi], [\psi']:~B \rightarrow Aut(K)/Int(K)$, then the equivalent extensions define the same maps. By (5) these maps are actually homomorphisms (unlike e.g.$\psi$). For a given $\psi$ if there is $\chi$ so that $(\psi,\chi)$ does define an extension of $B$ by $K$ we say that the extension is associated to (the homomorphism) $[\psi]$. That does not mean that any given homomorphism in $hom_{Group}(B,Aut(K)/Int(K))$ is associated to any extension, nor it means that if a homomorphism is associated to some extension, that every its representative in $hom_{Set}(B,Aut(K))$ is a part of a pair $(\psi,\chi)$ defining an extension. To see that situation in more detail we start with a given automorphism, which we call $\theta$ , and choose an element $\psi(a)\in\theta(a)$, the representative of a coset in $Aut(K)/Int(K)$; that choice should be specified for all $a \in B$. Note that for any $\rho \in Aut(K), a \in K$ we have, by direct inspection, $\rho Ad_K(a)\rho^{-1} = Ad_K(\rho(a))$. Thus there is a well-defined function
so choosing $\psi(a b) \in [\psi(a b)]$ is the same as choosing it in $[\psi(a)][\psi(b)]$ and guarantees that $\psi(a b)^{-1}\psi(a)\psi(b)$ is in $Int(K)$. Let us choose some $h$ so that $Ad_K \circ h$ is interpretable as a genuine composition.
what is by associativity the same as
Thus $Ad_K(h(a b,c)\psi(c)^{-1}h(a,b)) = Ad_K(h(a,b c)h(b,c)).$ Two elements of $K$ generate the same automorphism iff they differ by a central element. Thus
for a unique central element $z(a,b,c) \in Z(K).$ The correspondence $z : (a,b,c) \mapsto z(a,b,c)$ maps $B \times B \times B$ into $Z(K)$.
$z$ is an (Abelian) 3-cocycle with values in $_/Z(K)_{\psi^{-1}}$ ($Z(K)$ understood as trivial-$\psi^{-1}$ $B$-bimodule):
To see this we calcuate
Compare
(i) If we choose a different $h$ such that
then $z$ will change only up to a 3-coboundary $d f,$ i.e. there is a function $f : B \times B \rightarrow Z(K)$, such that $z' = (d f)z$ where
(ii) Conversely, if $z$ is a 3-cocycle obtained from $\psi$ as above and $d f$ is a 3-coboundary, then there is a $h'$ determining the same inner automoprhism of $K$ such that the corresponding 3-cocycle $z'$ is equal to $(df)z$.
(iii) Let $\psi, \psi' : B \rightarrow Aut(K)$ be two set-theoretic sections so that $[\psi] = [\psi'] = \theta : B \rightarrow Aut(K)/Int(K)$, then (for arbitrary choice of $h$, $h'$) the cocycles $z$ and $z'$ obtained as above differ only up to a 3-coboundary. $\|$
(i) Choose two different $h',h: B \times B \rightarrow K$ such that $Ad_K(h') = Ad_K(h)$. Then $h'(a,b) = h(a,b)f(a,b)$ where $f : B \times B \rightarrow Z(K)$ is some function with values in center of $K$. A direct comparison of (9) written for $h,z$ and $h',z'$ respectively proves the assertion.
(ii) Trivial: Any $f : B \times B \rightarrow Z(K)$ such that $h' = hf$ will not change the inner automorphism. Thus any central 3-coboundary $df$ can be obtained by changing a choice of $h$.
(iii) $[\psi'] = [\psi]$ implies that exists $k : B \rightarrow K, \psi'(a) = \psi(a)Ad_K(k(a)).$ Then
Thus $h'(a,b) = k(a b)^{-1}h(a,b)[\psi(b)^{-1}k(a)]k(b),$ for appropriate choice of $h'$ - what can change $z'$ up to coboundary - using the freedom from (i). If we want formula involving $\psi'$ instead than we use $\psi'(a) = \psi(a)Ad_K(k(a))$ to obtain $k(a b)h'(a,b) = h(a,b)k(b)[\psi'(b)^{-1}k(a)]$. Using that and previous identities,
for all $a,b,c \in B$. Thus $h'(a b,c)\psi'(c)^{-1}h'(a,b) = h'(a,b c)h'(b,c)z(a,b,c)$ i.e. our choice of $h'$ insured no change in $z$. Of course that means that in arbitrary choice of $h'$ we do not miss more than a coboundary by (i).
A given homomorphism $\theta : B \times B \rightarrow Aut(K)/Int(K)$ is associated to some extension of $B$ by $K$ iff $z$ is a 3-coboundary.
Indeed, if $\theta$ is associated to an extension, then we know that there is an isomorphism of the extension with a cross product given by some cocycle $\chi$ and some automorphism $\psi$ such that $[\psi] = \theta$. But using the identification, $\chi = h$ for that particular choice of $\psi$, so that $z = 1$. By the proposition, every other $z$ obtained from $\theta$ is in the same cohomology class, thus every such $z$ is a coboundary. Conversely, if $z$ is a coboundary, then by the proposition, we can change it to $z = 1$, and then we have all the conditions for a cross product extension satisfied.
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
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
the real numbers as an extension of the circle group
the spin group as an extension of the special orthogonal group
etc.
group extension, ∞-group extension
Original articles:
Samuel Eilenberg, Saunders MacLane, Group Extensions and Homology, Annals of Mathematics 43 4 (1942) 757-831 [doi:10.2307/1968966, jstor:1968966]
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:1969174
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 accounts
A. G. Kurosh, Theory of groups
Kenneth Brown, Cohomology of Groups, Graduate Texts in Mathematics, 87, Springer 1982 (doi:10.1007/978-1-4684-9327-6)
Lecture notes and similar include
Brian Conrad, Group cohomology and group extensions (pdf)
Patrick Morandi, Group extensions and $H^3$ (pdf)
Nobabelian cohomology (pdf)
Raphael Ho, Classifications of group extensions and $H^2$ (pdf)
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, 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.
A theory for central 2-group extensions is here:
See also references to Dedecker listed here.
A bit of discussion of some occurences of central extensions of groups in physics is in
(In fact there are many more than mentioned in that introduction.)
Extensions of supergroups are discussed in
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