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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{group extension} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory} [[!include group theory - contents]] \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{PropertiesGeneral}{General}\dotfill \pageref*{PropertiesGeneral} \linebreak \noindent\hyperlink{fibers_of_extensions_are_normal_subgroups}{Fibers of extensions are normal subgroups}\dotfill \pageref*{fibers_of_extensions_are_normal_subgroups} \linebreak \noindent\hyperlink{Torsors}{Extensions as torsors / principal bundles}\dotfill \pageref*{Torsors} \linebreak \noindent\hyperlink{SplitExtensionsAndSemidirectProductGroups}{Split extensions and semidirect product groups}\dotfill \pageref*{SplitExtensionsAndSemidirectProductGroups} \linebreak \noindent\hyperlink{PropertiesCentralGroupExtensions}{Central group extensions}\dotfill \pageref*{PropertiesCentralGroupExtensions} \linebreak \noindent\hyperlink{CentralExtensionClassificationByGroupCohomology}{Classification by group cohomology}\dotfill \pageref*{CentralExtensionClassificationByGroupCohomology} \linebreak \noindent\hyperlink{FormulationInHomotopyTheory}{Formulation in homotopy theory}\dotfill \pageref*{FormulationInHomotopyTheory} \linebreak \noindent\hyperlink{PropertiesAbelianGroupExtensions}{Abelian group extensions}\dotfill \pageref*{PropertiesAbelianGroupExtensions} \linebreak \noindent\hyperlink{SchreierTheory}{Nonabelian group extensions (Schreier theory)}\dotfill \pageref*{SchreierTheory} \linebreak \noindent\hyperlink{SchreierTheoryTraditional}{Traditional description}\dotfill \pageref*{SchreierTheoryTraditional} \linebreak \noindent\hyperlink{2Coboundaries}{Comparing different extensions; 2-coboundaries}\dotfill \pageref*{2Coboundaries} \linebreak \noindent\hyperlink{SchreierTheorynPOV}{Formulation in homotopy theory}\dotfill \pageref*{SchreierTheorynPOV} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak \noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{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]], $A \to \hat G \to G$. \hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition} \begin{defn} \label{GroupExtension}\hypertarget{GroupExtension}{} Two consecutive [[homomorphisms]] of [[groups]] \begin{equation} A \overset{i}\hookrightarrow \hat G\overset{p}\to G \label{shortExtension}\end{equation} are a \textbf{[[short exact sequence]]} if \begin{enumerate}% \item $i$ is [[monomorphism]], \item $p$ an [[epimorphism]] \item the [[image]] of $i$ is all of the [[kernel]] of $p$: $ker(p)\simeq im(i)$. \end{enumerate} We say that such a short exact sequence exhibits $\hat G$ as an \textbf{extension} of $G$ by $A$. If $A \hookrightarrow \hat G$ factors through the [[center]] of $\hat G$ we say that this is a \textbf{[[central extension]]}. \end{defn} \begin{remark} \label{}\hypertarget{}{} Sometimes in the literature one sees $\hat G$ called an extension ``of $A$ by $G$''. This is however in conflict with terms such as \emph{[[central extension]]}, \emph{[[extension of principal bundles]]}, etc, where the extension is always regarded \emph{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.) \end{remark} 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. \begin{defn} \label{}\hypertarget{}{} A sequence $A \to \hat G \to G$ is a [[short exact sequence]] of groups precisely if the [[delooping]] \begin{displaymath} \mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G \end{displaymath} is a [[fiber sequence]] in the [[(2,1)-category]] [[Grpd]]. \end{defn} This says that group extensions are special cases of the general notion discussed at \emph{[[∞-group extension]]}. See there for more details. \begin{defn} \label{MorphismOfGroupExtensions}\hypertarget{MorphismOfGroupExtensions}{} 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]] \begin{displaymath} \itexarray{ && \hat G_1 \\ & \nearrow && \searrow \\ A &&\downarrow^{\mathrlap{f}}&& G \\ & \searrow && \nearrow \\ && \hat G_2 } \,. \end{displaymath} \end{defn} \begin{prop} \label{}\hypertarget{}{} A morphism of extensions as in def. \ref{MorphismOfGroupExtensions} is necessarily an [[isomorphism]]. \begin{equation} \itexarray{ 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 } \,. \label{equivExt}\end{equation} \end{prop} \begin{proof} By the [[short five lemma]]. \end{proof} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} We discuss properties of group extensions in stages, \begin{itemize}% \item \hyperlink{PropertiesGeneral}{General properties} \item \hyperlink{PropertiesCentralGroupExtensions}{Central group extensions} \item \hyperlink{PropertiesAbelianGroupExtensions}{Abelian group extensions} \item \hyperlink{SchreierTheory}{Schreier theory for nonabelian extensions} \end{itemize} \hypertarget{PropertiesGeneral}{}\subsubsection*{{General}}\label{PropertiesGeneral} \hypertarget{fibers_of_extensions_are_normal_subgroups}{}\paragraph*{{Fibers of extensions are normal subgroups}}\label{fibers_of_extensions_are_normal_subgroups} \begin{prop} \label{}\hypertarget{}{} For $A \hookrightarrow \hat G \to G$ a group extension, the inclusion $A \hookrightarrow \hat G$ is a [[normal subgroup]] inclusion. \end{prop} \begin{proof} 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 \begin{displaymath} \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} \end{displaymath} and hence $g a g^{-1}$ is in the [[kernel]] of $p$. By the defining exactness property therefore it is in the [[image]] of $i$. \end{proof} \hypertarget{Torsors}{}\paragraph*{{Extensions as torsors / principal bundles}}\label{Torsors} \begin{prop} \label{}\hypertarget{}{} 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 \begin{displaymath} \rho : A \times \hat G \stackrel{(i,Id)}{\to} \hat G \times_G \hat G \stackrel{\cdot}{\to} \hat G \,. \end{displaymath} \end{prop} \begin{proof} That $\rho$ is indeed an action \emph{over} $B$ in that \begin{displaymath} \itexarray{ A \times \hat G &&\stackrel{\rho}{\to}&& \hat G \\ & {}_{\mathllap{ p \circ p_2}}\searrow && \swarrow_{\mathrlap{p}} \\ && G } \end{displaymath} follows from the fact that $p$ is a group homomorphism and that $A$ is in its [[kernel]]. That $A$ is actually \emph{equal} to the kernel gives the principality condition \begin{displaymath} (\rho, p_2) : A\times \hat G \stackrel{\simeq}{\to} \hat G \times_G \hat G \,. \end{displaymath} \end{proof} 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 \begin{displaymath} 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 \end{displaymath} 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]]. \hypertarget{SplitExtensionsAndSemidirectProductGroups}{}\paragraph*{{Split extensions and semidirect product groups}}\label{SplitExtensionsAndSemidirectProductGroups} \begin{defn} \label{SplitExtension}\hypertarget{SplitExtension}{} A group extension $A \to \hat G \stackrel{p}{\to} G$ is called \textbf{split} if there is a [[section]] [[homomorphism]] $\sigma \colon G \to \hat G$, hence a group homomorphism such that $p \circ \sigma = id_G$. \end{defn} \begin{remark} \label{}\hypertarget{}{} 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. \end{remark} \begin{prop} \label{SplitExtensionsAreSemidirectProducts}\hypertarget{SplitExtensionsAreSemidirectProducts}{} Split extensions $\hat G$ of $G$ by $A$, def. \ref{SplitExtension}, are equivalently [[semidirect product groups]] \begin{displaymath} A \hookrightarrow \hat G \simeq A \rtimes_\rho G \to G \end{displaymath} for some [[action]] $\rho \colon G \times A \to A$ of $G$ on $A$. \end{prop} 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 \begin{displaymath} (a_1, g_1) \cdot (a_2, g_2) = (a_1 \cdot \rho(g_1)(a_2) , g_1 \cdot g_2) \,. \end{displaymath} The inclusion of $A$ is by \begin{displaymath} a \mapsto (a,e) \end{displaymath} and the projection to $G$ is by \begin{displaymath} (a,g) \mapsto g \,. \end{displaymath} \begin{proof} 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 $A$: \begin{displaymath} \rho \colon A \times G \stackrel{(i,\sigma)}{\to} \hat G \times \hat G \stackrel{\rho_{ad}}{\to} \hat G \end{displaymath} \begin{displaymath} (a,g) \mapsto \sigma(g)^{-1} \cdot a \cdot \sigma(g) \,. \end{displaymath} Then (\ldots{}) \end{proof} \begin{prop} \label{}\hypertarget{}{} A split extension $A \to \hat G \to G$ is a [[central extension]] precisely if the [[action]] $\rho$ induced from it as in prop. \ref{SplitExtensionsAreSemidirectProducts} is trivial. \end{prop} \begin{proof} 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 \begin{displaymath} (a,e) \cdot (e,g) = (a, g) \end{displaymath} but \begin{displaymath} (e,g) \cdot (a,e) = (\rho(g)(a), g) \,. \end{displaymath} 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. \end{proof} \begin{prop} \label{}\hypertarget{}{} 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. \end{prop} \begin{prop} \label{}\hypertarget{}{} If we have a split extension the different splittings are given by [[derivation on a group|derivation]]s, 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-cocylces of $G$ with (twisted) coefficients in $A$, as considered in, for instance, Serre's notes on [[Galois cohomology]]. \end{prop} \hypertarget{PropertiesCentralGroupExtensions}{}\subsubsection*{{Central group extensions}}\label{PropertiesCentralGroupExtensions} We discuss properties of \emph{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 \emph{\hyperlink{SchreierTheory}{Nonabelian group extensions (Schreier theory)}} but is considerably less complex to write out in components. We first discuss the \begin{itemize}% \item \hyperlink{CentralExtensionClassificationByGroupCohomology}{Classification by group cohomology} \end{itemize} of central extensions in components, and then show in \begin{itemize}% \item \hyperlink{FormulationInHomotopyTheory}{Formulation in homotopy theory} \end{itemize} how this follows from a more systematic abstract theory. \hypertarget{CentralExtensionClassificationByGroupCohomology}{}\paragraph*{{Classification by group cohomology}}\label{CentralExtensionClassificationByGroupCohomology} We discuss the classification of \emph{central} extensions by [[group cohomology]]. This is a special case of the more general (and more complicated) discussion below in \emph{\hyperlink{SchreierTheory}{Nonabelian group extensions (Schreier theory)}}. For $G$ a [[group]] and $A$ an [[abelian group]], write \begin{displaymath} H^2_{grp}(G,A) \;\; \in Ab \end{displaymath} for the degree-2 [[group cohomology]] of $G$ with [[coefficients]] in $A$, and write \begin{displaymath} Ext(G,A) \;\;\in Ab \end{displaymath} for the group of central extensions of $G$ by $A$. \begin{theorem} \label{}\hypertarget{}{} There is a [[natural equivalence]] \begin{displaymath} Ext(G,A) \simeq H^2_{Grp}(G,A) \,. \end{displaymath} \end{theorem} We prove this below as prop. \ref{ExtractionAndReconstructionConsituteEquivalence}. Here we first introduce stepwise the ingredients that go into the proof. \begin{defn} \label{CentralExtensionAssociatedTo2Cocycle}\hypertarget{CentralExtensionAssociatedTo2Cocycle}{} \textbf{(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 \begin{displaymath} G \times_c A \in Grp \end{displaymath} 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 \begin{displaymath} (g_1, a_1) \cdot (g_2, a_2) \coloneqq (g_1 \cdot g_2 ,\; a_1 + a_2 + c(g_1, g_2)) \,. \end{displaymath} \end{defn} \begin{prop} \label{}\hypertarget{}{} 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 \begin{displaymath} Rec : H^2_{Grp}(G,A) \to Ext(G,A) \,. \end{displaymath} \end{prop} \begin{proof} Forming the product of three elements of $G \times_c A$ bracketed to the left is, according to def. \ref{CentralExtensionAssociatedTo2Cocycle}, \begin{displaymath} \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) \,. \end{displaymath} Bracketing the same three elements to the right yields \begin{displaymath} \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) \,. \end{displaymath} The difference between the two expressions is read off to be precisely \begin{displaymath} (1, (d c) (g_1, g_2, g_3)) \,, \end{displaymath} 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 \begin{displaymath} (g,a) \cdot (g^{-1}, - a - c(g,g^{-1})) = (e, a - a - c(g,g^{-1})+ c(g,g^{-1}) ) \end{displaymath} 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 \href{group+cohomology#Degree2}{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 \begin{displaymath} i \colon A \to G \times_c A \end{displaymath} 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 \begin{displaymath} A \stackrel{i}{\hookrightarrow} G \times_c A \stackrel{p}{\to} G \end{displaymath} is indeed a group extension. It is a [[central extension]] again using the assumption that $c$ is normalized $c(g,e) = c(e,g) = 0$: \begin{displaymath} (g,a) \cdot (e,\tilde a) = (g, a + \tilde a + 0) = (e,\tilde a) \cdot (g,a) \,. \end{displaymath} 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 \begin{displaymath} \tilde c (g_1,g_2) \coloneqq c(g_1,g_2) - h(g_1) - h(g_2) + h(g_1 g_2) \,. \end{displaymath} We claim that then we have a homomorphism of central extensions (hence an isomorphism) of the form \begin{displaymath} \itexarray{ 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 } \,. \end{displaymath} To see this we check for all elements that \begin{displaymath} \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} \,. \end{displaymath} Hence the construction of $G \times_c A$ indeed defines a function $H^2_{Grp}(G,A) \to CentrExt(G,A)$. \end{proof} Assume the [[axiom of choice]] in the ambient [[foundations]]. \begin{defn} \label{2CocycleExtractedFromCentralExtension}\hypertarget{2CocycleExtractedFromCentralExtension}{} \textbf{(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 \begin{displaymath} c \colon (g_1, g_2) \mapsto -\sigma(g_1)^{-1} \sigma(g_2)^{-1} \sigma(g_1 g_2) \in A \,, \end{displaymath} where on the right we are using that by the section-property of $\sigma$ and the group homomorphism property of $p$ \begin{displaymath} p(\sigma(g_1)^{-1} \sigma(g_2)^{-1} \sigma(g_1 g_2)) = 1 \end{displaymath} and hence by the exactness of the extension the argument is in $A \hookrightarrow \hat G$. \end{defn} Below in remark \ref{RelationBetweenComponentCocycleExtractionAndHomotopyTheory} is a discussion of how this construction arises from a more systematic discussion in [[homotopy theory]]. \begin{prop} \label{}\hypertarget{}{} The construction of prop. \ref{2CocycleExtractedFromCentralExtension} indeed yields a 2-cocycle in [[group cohomology]]. It extends to a morphism \begin{displaymath} Extr \colon Ext(G,A) \to H^2_{Grp}(G,A) \,. \end{displaymath} \end{prop} \begin{proof} The cocycle condition to be checked is that \begin{displaymath} 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 \end{displaymath} for all $g_0, g_1, g_2 \in G$. Writing this out with def. \ref{2CocycleExtractedFromCentralExtension} yields \begin{displaymath} \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} \,. \end{displaymath} 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 \begin{displaymath} h \colon g \mapsto \tilde \sigma(g)\sigma(g)^{-1} \,, \end{displaymath} 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 $G$. \end{proof} \begin{prop} \label{ExtractionAndReconstructionConsituteEquivalence}\hypertarget{ExtractionAndReconstructionConsituteEquivalence}{} The two morphisms of def. \ref{CentralExtensionAssociatedTo2Cocycle} and def. \ref{2CocycleExtractedFromCentralExtension} exhibit the [[equivalence]] \begin{displaymath} H^2_{Grp}(G,A) \stackrel{\underoverset{\simeq}{Extr}{\leftarrow}}{\underset{Rec}{\to}} CentrExt(G,A) \,. \end{displaymath} \end{prop} \begin{proof} 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 \begin{displaymath} \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} \,, \end{displaymath} 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. \end{proof} \hypertarget{FormulationInHomotopyTheory}{}\paragraph*{{Formulation in homotopy theory}}\label{FormulationInHomotopyTheory} 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 \emph{[[looping and delooping]]}), complementing the \href{CentralExtensionClassificationByGroupCohomology }{above} component-wise discussion. Let \begin{displaymath} A \hookrightarrow \hat G \to G \end{displaymath} be a central group extension, def. \ref{GroupExtension}, hence with $A$ an [[abelian group]] included in the [[center]] of $G$. Then $A$ is in particular a [[normal subgroup]] and hence the homorphism \begin{displaymath} (A \to \hat G) \end{displaymath} 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 $A$ \begin{displaymath} (A \to \hat G) = \{ g \stackrel{a}{\to} a \cdot g \} \,. \end{displaymath} \begin{defn} \label{}\hypertarget{}{} Write \begin{displaymath} \mathbf{B}(A \to \hat G) \in Grpd \end{displaymath} 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-[[coskeleton|coskeletal]]) [[Kan complex]] that realizes this as a [[truncated object of an (infinity,1)-category|2-truncated]] [[∞-groupoid]]. \begin{displaymath} \mathbf{B}(A \to \hat G) \in \infty Grpd \,. \end{displaymath} \end{defn} \begin{prop} \label{}\hypertarget{}{} The obvious strict [[2-functor]] \begin{displaymath} \mathbf{B}(A \to \hat G) \stackrel{\simeq}{\to} \mathbf{B}H \end{displaymath} is an [[weak homotopy equivalence|equivalence]] of 2-groupoids. \end{prop} \begin{proof} 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]] \begin{displaymath} N\mathbf{B}(A \to \hat G) \to N \mathbf{B}H \end{displaymath} is an acyclic [[Kan fibration]], hence a [[weak equivalence]] in the standard [[model structure on simplicial sets]]. \end{proof} \begin{prop} \label{}\hypertarget{}{} The extension $A \to \hat G \to G$ sits in a long [[homotopy fiber sequence]] in the [[(∞,1)-category]] [[∞Grpd]] of the form \begin{displaymath} 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 \end{displaymath} which in [[Kan complexes]]/[[simplicial sets]] is [[presentable (infinity,1)-category|presented]] by the [[zigzag]] of [[n-functors]] between [[strict ∞-groupoid]] (sequence of [[2-anafunctors]]) of the form \begin{displaymath} \itexarray{ && && && && && \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 } \,. \end{displaymath} In particular, the induced [[connecting homomorphism]] \begin{displaymath} \mathbf{c} : \mathbf{B}G \to \mathbf{B}^2 A \end{displaymath} is the [[group cohomology]] [[cocycle]] that classifies the delooped extension as a $\mathbf{B}A$-[[principal 2-bundle]]. \end{prop} \begin{proof} 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. \end{proof} \begin{remark} \label{RelationBetweenComponentCocycleExtractionAndHomotopyTheory}\hypertarget{RelationBetweenComponentCocycleExtractionAndHomotopyTheory}{} The construction in def. \ref{2CocycleExtractedFromCentralExtension} \begin{displaymath} Rec : CentrExt(G,A) \to H^2_{Grp}(G,A) \end{displaymath} is precisely the result of moving set-theoretically through the [[zigzag]] \begin{displaymath} \itexarray{ \mathbf{B}(A \to \hat G) &\to& \mathbf{B}(A \to 1) = \mathbf{B}^2 A \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}G } \end{displaymath} 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 \emph{[[mapping cone]]} in the section \emph{\href{mapping%20cone#HomologyExactSequencesAndFiberSequences}{Homology exact sequences and fiber sequences}}. \end{remark} \hypertarget{PropertiesAbelianGroupExtensions}{}\subsubsection*{{Abelian group extensions}}\label{PropertiesAbelianGroupExtensions} \begin{remark} \label{}\hypertarget{}{} For $A, G \in$ [[Ab]] $\hookrightarrow$ [[Grp]] even a [[central extension]] $\hat G$ of $G$ by $A$ is not necessarily itself an abelian group. \end{remark} But by prop. \ref{ExtractionAndReconstructionConsituteEquivalence} above it is so if the group 2-cocycle that classifies the extension is symmetric: \begin{defn} \label{}\hypertarget{}{} A 2-cocycle $c \colon G \times G \to A$ in [[group cohomology]] is \textbf{symmetric} if \begin{displaymath} \forall_{g_1, g_2 \in G} c(g_1, g_2) = c(g_2, g_1) \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} 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. \end{remark} \begin{defn} \label{}\hypertarget{}{} Write \begin{displaymath} H^2_{Grp}(G,A)_{sym} \hookrightarrow H^2_{Grp}(G,A) \end{displaymath} for the set (group) of classes of symmetric group 2-cocycles on $G$ with coefficients in $A$. \end{defn} \begin{defn} \label{}\hypertarget{}{} For $G,A \in Ab \hookrightarrow Grp$, write $Ext(G,A)$ for the subset of equivalence class of abelian group extensions of $G$ by $A$. \end{defn} The theory of \emph{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 \emph{[[projective resolution]]} the section \begin{itemize}% \item \emph{\href{projective+resolution#ProjectiveResolutionsForGroupCocycles}{Projective resolutions adapted to group cocycles}} \end{itemize} and \begin{itemize}% \item \emph{\href{projective+resolution#DerivedHomFunctor}{Derived Hom-functor / Ext-functor}} . \end{itemize} \hypertarget{SchreierTheory}{}\subsubsection*{{Nonabelian group extensions (Schreier theory)}}\label{SchreierTheory} We discuss the classification theory for the general case of nonabelian group extensions, first in the form of \begin{itemize}% \item \emph{\href{SchreierTheoryTraditional}{Traditional Schreier theory}} \end{itemize} and then more abstractly in the language of [[homotopy theory]] in \begin{itemize}% \item \emph{\hyperlink{SchreierTheorynPOV}{Homotopy theory perspective on Schreier theory}} \end{itemize} \hypertarget{SchreierTheoryTraditional}{}\paragraph*{{Traditional description}}\label{SchreierTheoryTraditional} [[Otto Schreier]] (1926) and [[Samuel Eilenberg|Eilenberg]]-[[Saunders MacLane|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 \textbf{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 $p:G\to B$. \textbf{Note.} The exposition which follows in this long ``traditional'' section of this entry is mainly from personal notes of Zoran \v{S}koda from 1997. Each element $g$ of $G$ defines an inner automorphism $\phi(g)$ of $K$ by $\phi(g)(k) = gkg^{-1}$. 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 $\phi(g_1k') = \phi(g_2kk') = \phi(g_2)\phi(kk')$ 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 \begin{displaymath} \psi \stackrel{def}{=} \phi \circ \sigma: B \rightarrow Aut(K). \end{displaymath} \textbf{Warning.} Unlike $\phi$, the map $\psi$ is \emph{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 \begin{displaymath} B\times K\ni (b,k)\mapsto\sigma(b)k \in G, \end{displaymath} 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, \begin{displaymath} 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. \end{displaymath} Now $p(\sigma(b_1)\sigma(b_2)) = p(b_1b_2)$ so \begin{displaymath} \chi(b_1,b_2) \stackrel{def}{=} \sigma(b_1b_2)^{-1}\sigma(b_1)\sigma(b_2) \in K. \end{displaymath} This formula clearly defines a function $\chi : B \times B \rightarrow K$. In this notation, \begin{displaymath} \itexarray{ 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. } \end{displaymath} and using bijection of $G$ with $B\times K$ this can be expressed in terms of elements in $B\times K$ so that \begin{equation} (b_1,k_1)(b_2,k_2) = (b_1b_2,\chi(b_1,b_2)[\psi(b_2)^{-1}(k_1)] k_2). \label{mrule}\end{equation} According to this formula, \emph{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, \emph{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 \begin{displaymath} [(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))). \end{displaymath} From the other side, this has to be the same, by associativity, to \begin{displaymath} (a,e)[(b,e)(c,e)] = (a,e)(bc,\chi(b,c)) = (a b c,\chi(a,b c)\chi(b,c)) \end{displaymath} where we took into account that expressions like $[\psi^{-1}(b)(e)] = e$, because $\psi(b)$ is an \emph{automorphism} for each $b \in B$. Comparing the expressions above we obtain \begin{equation} \chi(a b,c)\psi(c)^{-1}(\chi(a,b)) = \chi(a,b c)\chi(b,c), for all a,b,c \in B. \label{psi1}\end{equation} If the pair $(\chi,\psi)$ is constructed as above, then \begin{displaymath} \itexarray{ \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, } \end{displaymath} where $Ad_K$ is the canonical map $K \rightarrow Int(K)$, $k\mapsto k(-)k^{-1}$. Thus we obtain the relation \begin{equation} \psi(a)\psi(b) = \psi(a b) Ad_K(\chi(a,b)) \label{psi2}\end{equation} \begin{defn} \label{}\hypertarget{}{} Let $B$ and $K$ be two groups. Let $\chi: B \times B \rightarrow K$ and $\psi : B \rightarrow Aut(K)$ satisfy \eqref{psi1} and \eqref{psi2}. 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 \textbf{nonabelian group 2-cocycle with automorphisms}, and the family $\{\psi(b) | b \in B \}$ -- a system of automorphisms \end{defn} A 2-cocycle $\chi$ is \textbf{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. \eqref{psi2}). 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 \eqref{psi1} becomes \begin{displaymath} \chi(a b,c)\chi(a,b) = \chi(a,b c)\chi(b,c). \end{displaymath} \begin{theorem} \label{}\hypertarget{}{} If formulas \eqref{psi1} and \eqref{psi2} are both satisfied, then the formula \eqref{mrule} for multiplication of pairs defines a group multiplication on $B \times K$. That set, together with multiplication \eqref{mrule} is called the \textbf{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 \textbf{(external) semidirect product}. \end{theorem} \begin{proof} 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 \eqref{psi1} and \eqref{psi2} it can be done. \begin{displaymath} \itexarray{ \psi(a)\psi(e)k & =& \psi(a)Ad_K(\chi(a,e))k \\ & = &\psi(a)\chi(a,e)k\chi(a,e)^{-1} } \end{displaymath} where we used \eqref{psi2}. Thus $Ad_K(\chi(a,e)) = \psi(e)$ and therefore it does not depend on $a$. Then use \eqref{psi1} 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 \emph{unit} element is given by $(e, \chi(e,e)^{-1})$. To verify that it is also a right unit we compute \begin{displaymath} \itexarray{ (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}) } \end{displaymath} what is equal to $(a,b)$ by just proved statement that $\chi(a,e)$ does not depend on $a$. Now use \eqref{psi1} with $a = b = e$ to get \begin{equation} \psi(c)^{-1}(\chi(e,e)) = \chi(e,c), \forall c \in B. \label{psiac}\end{equation} Thus we can verify that $(e, \chi(e,e)^{-1})$ is a left unit too by a calculation as follows. Namely \begin{displaymath} (e,\chi(e,e)^{-1})(a,b)= (a,\chi(e,a)\psi(a)^{-1}(\chi(e,e)^{-1})b) \end{displaymath} by the definition of the product. Then by \eqref{psiac}, this equals to \begin{displaymath} (a,\psi(a)^{-1}(\chi(e,e))\psi(a)^{-1}(\chi(e,e)^{-1})b) \end{displaymath} 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 \begin{displaymath} (a^{-1},\chi(a,a)^{-1}\chi(e,e)^{-1}). \end{displaymath} To this aim, we calculate \begin{displaymath} (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}), \end{displaymath} because $\psi(a)^{-1}(e) = e$. Furthermore, \begin{displaymath} (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}), \end{displaymath} because $\psi(a)^{-1}(e) = e$. Next, \begin{displaymath} (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})) \end{displaymath} what equals $(e,\chi(e,e)^{-1})$. Indeed, \eqref{psi1} 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 \eqref{psiac} and take inverse of both sides to obtain \begin{displaymath} \psi(a)^{-1}(\chi(a,a^{-1})^{-1}\chi(e,e)^{-1})) = \chi(a^{-1},a)^{-1}\chi(a,e)^{-1}. \end{displaymath} 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. \end{proof} Given groups $K$ and $B$ and any maps $\chi$ and $\psi$ satisfying \eqref{psi1} and \eqref{psi2}, 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 \begin{displaymath} \itexarray{ \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 } \end{displaymath} for all $b_1,b_2 \in B$ for all $k \in K$ in all these lines. The last line is true by \eqref{psi1}. 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 \begin{displaymath} (e,\psi(b)k)(b,e) = (b, \chi(e,b)\psi(b)^{-1}(\chi(e,e)^{-1}\psi(b)(k))) = (b, k) \end{displaymath} by \eqref{psiac}. \begin{prop} \label{}\hypertarget{}{} The following are equivalent \begin{itemize}% \item (i) extension \eqref{shortExtension} is split \item (ii) for extension \eqref{shortExtension} there is a subgroup $B_1 \subset G$ such that $B_1 \cap i(K) = 1$ and $B_1i(K) = G$ ($G$ is an internal semidirect product of $K$ and $B_1$). \item (iii) extension \eqref{shortExtension} is isomorphic to an external semidirect product of $K$ and $B$. \end{itemize} \end{prop} \begin{proof} (i) $\Rightarrow$ (ii) If the extension is split then there is a \emph{homomorphism} $\sigma : B \rightarrow G$ such that $p \circ \sigma = id_B$. Let $B_1 = \sigma(B)$. By exactness of \eqref{shortExtension}), 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 \eqref{mrule} without a cocycle. The map $\sigma : B \rightarrow G$, $B \ni b \mapsto (1_K,b) \in K \times B$, splits the sequence. \end{proof} \begin{defn} \label{}\hypertarget{}{} An extension \eqref{shortExtension} is Abelian iff $K$ is Abelian. An Abelian extension \eqref{shortExtension} 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 \eqref{shortExtension} is Abelian iff $G$ is Abelian. \end{defn} Remarks. (i) Note that \eqref{psi2} 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 \eqref{shortExtension} 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). \hypertarget{2Coboundaries}{}\paragraph*{{Comparing different extensions; 2-coboundaries}}\label{2Coboundaries} 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 \eqref{equivExt}. We know that $\epsilon|_K : i(k) \stackrel{\epsilon}\mapsto i'(k)$, for all $k \in K$. The formula for $i$ in $\backslash$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 \begin{displaymath} \itexarray{ \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). } \end{displaymath} Now multiply more general elements in $G$: \begin{displaymath} \itexarray{ \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) } \end{displaymath} what should be the same as \begin{displaymath} \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) \end{displaymath} In a special case, when $k_1 = e_K$ we have therefore \begin{equation} \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) \label{equiv1}\end{equation} In order to obtain a relation between $\psi'(b)(k)$ and $\psi(b)(k)$ note that \begin{displaymath} \epsilon ((e,\chi(e,e)^{-1}k)(b,e)) = (e, \chi'(e,e)^{-1}k)(b,\lambda(b)). \end{displaymath} That is equivalent to any in the following chain of formulas: \begin{displaymath} \itexarray{ \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) } \end{displaymath} Then by \eqref{psiac}, it follows that \begin{displaymath} \itexarray{ \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) } \end{displaymath} Now invert the maps in $Aut(K)$ to obtain \begin{equation} \psi'(b) = \psi(b)Ad_K(\lambda(b)^{-1}) \label{equiv2}\end{equation} Thus we obtain \begin{theorem} \label{}\hypertarget{}{} 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 \eqref{equiv1} and \eqref{equiv2} are valid. \end{theorem} \begin{proof} If function $\lambda$ takes values in the center of $B$ then \eqref{equiv2} 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 \eqref{psi2} 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 \emph{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 \emph{given} automorphism, which we call $\theta$ , and \emph{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 \begin{displaymath} Ad_K \circ h : B \times B \rightarrow Int(K), \,\,\, (Ad_K\circ h)(a,b) := \psi(a b)^{-1}\psi(a)\psi(b) \end{displaymath} \begin{itemize}% \item indeed \end{itemize} \begin{displaymath} \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) \end{displaymath} 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. \begin{displaymath} \itexarray{ (\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)) } \end{displaymath} what is by associativity the same as \begin{displaymath} \itexarray{ \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)). } \end{displaymath} 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 \begin{equation} h(a b,c)\psi(c)^{-1}h(a,b) = h(a,b c)h(b,c)z(a,b,c) \label{2semicoc}\end{equation} 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 $Z(K)$. \end{proof} \begin{prop} \label{}\hypertarget{}{} $z$ is an (Abelian) 3-cocycle with values in $_/Z(K)_{\psi^{-1}}$ ($Z(K)$ understood as trivial-$\psi^{-1}$ $B$-bimodule): \begin{equation} 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) \label{3coc}\end{equation} \end{prop} \begin{proof} To see this we calcuate \begin{displaymath} \itexarray{ 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) } \end{displaymath} Compare \begin{displaymath} \itexarray{ 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) } \end{displaymath} \end{proof} \begin{prop} \label{}\hypertarget{}{} (i) If we choose a different $h$ such that \begin{displaymath} Ad_K(h(a,b)) = \psi(a b)^{-1}\psi(a)\psi(b), \end{displaymath} 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 \begin{displaymath} (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. \end{displaymath} (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. $\|$ \end{prop} \begin{proof} (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 \eqref{2semicoc} 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 \begin{displaymath} \itexarray{ \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)). } \end{displaymath} 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, \begin{displaymath} \itexarray{ 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) } \end{displaymath} 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). \end{proof} \begin{cor} \label{}\hypertarget{}{} 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. \end{cor} \begin{proof} 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. \end{proof} \hypertarget{SchreierTheorynPOV}{}\paragraph*{{Formulation in homotopy theory}}\label{SchreierTheorynPOV} 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 \begin{itemize}% \item [[group cohomology]] \item [[nonabelian group cohomology]]. \end{itemize} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} By the above classification theorems, all the examples at \emph{[[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 \begin{itemize}% \item the [[real numbers]] as an extension of the [[circle group]] \begin{displaymath} \mathbb{Z} \to \mathbb{R} \to U(1) \,. \end{displaymath} \item the [[spin group]] as an extension of the [[special orthogonal group]] \begin{displaymath} \mathbb{Z}_2 \to Spin \to SO \end{displaymath} \item etc. \item [[universal central extension]] \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[group cohomology]] \begin{itemize}% \item [[Galois cohomology]] \item [[nonabelian group cohomology]], [[groupoid cohomology]] \end{itemize} \item \textbf{group extension}, [[∞-group extension]] [[Ext-group]], [[central extension]], [[maximal central extension]] \begin{itemize}% \item [[Baer sum]] \item [[ring extension]] \end{itemize} \item [[Lie group cohomology]] \begin{itemize}% \item [[∞-Lie groupoid cohomology]] \end{itemize} \item [[central extension of groupoids]] \item [[Lie algebra extension]] \item [[central charge]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general_2}{}\subsubsection*{{General}}\label{general_2} \begin{itemize}% \item [[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 \href{http://www.jstor.org/pss/1969174}{jstor} \item [[Saunders MacLane]], Cohomology theory in abstract groups. III. Operator homomorphisms of kernels. Ann. of Math. (2) 50, (1949). 736--761. \item [[Lawrence Breen]], Th\'e{}orie de Schreier sup\'e{}rieure, Ann. Sci. \'E{}cole Norm. Sup. (4) 25 (1992), no. 5, 465--514 \href{http://www.numdam.org/item?id=ASENS_1992_4_25_5_465_0}{numdam}. \end{itemize} Textbooks include \begin{itemize}% \item A. G. Kurosh, Theory of groups \item Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, \textbf{87}, Springer-Verlag, New York-Berlin, 1982. \end{itemize} Lecture notes and similar include \begin{itemize}% \item [[Brian Conrad]], \emph{Group cohomology and group extensions} (\href{http://math.stanford.edu/~conrad/249BPage/handouts/gpext.pdf}{pdf}) \item [[Terry Tao]], \emph{Some notes on group extensions} (\href{http://terrytao.wordpress.com/2010/01/23/some-notes-on-group-extensions/}{blog}) \item [[Patrick Morandi]], \emph{Group extensions and $H^3$} (\href{http://sierra.nmsu.edu/morandi/notes/GroupExtensions.pdf}{pdf}) \emph{Nobabelian cohomology} (\href{http://sierra.nmsu.edu/morandi/notes/nonabeliancohomology.pdf}{pdf}) \item Raphael Ho, \emph{Classifications of group extensions and $H^2$} (\href{http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Ho.pdf}{pdf}) \end{itemize} See also: \begin{itemize}% \item [[R. Brown]], [[T. Porter]], \emph{On the Schreier theory of non-abelian extensions: generalisations and computations}, Proc. Roy. Irish Acad. Sect. A, 96 (1996), 213 -- 227. \item [[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. \item [[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. \end{itemize} A theory for central 2-group extensions is here: \begin{itemize}% \item [[Antonio R. Garzón]] and E.M. Vitale, On the second cohomology categorical group and a Hochschild-Serre 2-exact sequence, Theory and Applications of Categories, Vol. 30 (2015), 933-984. (\href{http://www.tac.mta.ca/tac/volumes/30/27/30-27.pdf}{pdf}) \end{itemize} See also references to Dedecker listed [[zoranskoda:Paul Dedecker|here]]. \hypertarget{applications}{}\subsubsection*{{Applications}}\label{applications} A bit of discussion of some occurences of central extensions of groups in [[physics]] is in \begin{itemize}% \item G. Tuynman and W. Wiegerinck, \emph{Central extensions of physics} (\href{http://math.univ-lille1.fr/~gmt/PaperFolder/CentralExtensions.pdf}{pdf}) \end{itemize} (In fact there are many more than mentioned in that introduction.) Extensions of [[supergroups]] are discussed in \begin{itemize}% \item [[Christopher Drupieski]], \emph{Strict polynomial superfunctors and universal extension classes for algebraic supergroups} (\href{http://arxiv.org/abs/1408.5764}{arXiv:1408.5764}) \end{itemize} [[!redirects Schreier's theory]] [[!redirects Schreier theory]] [[!redirects extension of groups]] [[!redirects extensions of groups]] [[!redirects central extension of groups]] [[!redirects central extensions of groups]] [[!redirects group extension]] [[!redirects group extensions]] [[!redirects abelian group extension]] [[!redirects abelian group extensions]] \end{document}