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Group extension

In group theory, a group extension provides a method for constructing a new group G from a given normal subgroup N and a quotient group H, formalized by a short exact sequence $1 \to N \to G \to H \to 1, where the map from N to G is injective and the map from G to H is surjective with kernel N.^[1] This structure captures how G "extends" H by incorporating N as a normal subgroup, allowing the study of more complex groups in terms of simpler building blocks.^[2] Group extensions are fundamental in classifying groups and understanding their properties, such as solvability and nilpotency, as solvable groups can be built as towers of abelian extensions, while nilpotent groups arise from iterated central extensions where N lies in the center of G.^[3] Key types include split extensions, which correspond to semidirect products when there exists a homomorphism from H to G that splits the sequence, and central extensions, which play a crucial role in cohomology theory for determining equivalence classes of extensions.^[1] The classification of all inequivalent extensions of H by N—up to isomorphism preserving the sequence—forms the extension problem, often resolved using group cohomology; for instance, when N is an abelian H-module, the inequivalent extensions are parametrized by the second cohomology group H^2(H, N).^[4] Notable examples encompass direct products as trivial split extensions and non-split cases like the quaternion group as a central extension of the Klein four-group by \mathbb{Z}/2\mathbb{Z}, illustrating how extensions reveal symmetries in geometry, physics, and algebra.^[5] These constructions extend to broader contexts, such as abelian categories in homological algebra, underscoring their role in modern mathematics.^[2]

Basic Concepts

Definition

In group theory, a group extension provides a framework for constructing a larger group from two given groups, where one serves as a normal subgroup and the other as the quotient group. Formally, an extension of a group Q by a group N is specified by a short exact sequence of group homomorphisms

$1 \to N \xrightarrow{i} E \xrightarrow{p} Q \to 1,

where i is injective (embedding N into E as a subgroup), p is surjective (covering all of Q), and the sequence is exact, meaning \ker p = \operatorname{im} i and this image is normal in E.^[6] This structure implies that E has a normal subgroup isomorphic to N, and the quotient E / (\operatorname{im} i) \cong Q.^[6] The short exact sequence captures the essential group-theoretic relationships: the injectivity of i ensures no nontrivial elements of N are identified in E, the surjectivity of p guarantees that Q is fully realized as a quotient, and exactness at E confirms that the embedded copy of N precisely accounts for the kernel of the projection to Q.^[7] Here, group homomorphisms are functions f: G \to H between groups that preserve the group operation (f(gh) = f(g)f(h)), and the kernel of a homomorphism f is the normal subgroup \ker f = \{ g \in G \mid f(g) = e_H \}, where e_H is the identity in H.^[6] A fundamental example of such an extension is the direct product E = N \times Q, where i(n) = (n, e_Q) for n \in N and p(n, q) = q for (n, q) \in E, yielding a trivial extension in which elements of N and Q commute.^[6] Another representative case is the semidirect product E = N \rtimes_\phi Q, induced by a homomorphism \phi: Q \to \operatorname{Aut}(N) specifying an action of Q on N; the multiplication in E is defined by (n_1, q_1)(n_2, q_2) = (n_1 \phi_{q_1}(n_2), q_1 q_2), with i(n) = (n, e_Q) and p(n, q) = q.^[6] For instance, both \mathbb{Z}/6\mathbb{Z} and the symmetric group S_3 realize extensions of \mathbb{Z}/2\mathbb{Z} by \mathbb{Z}/3\mathbb{Z}, with the former being the direct product \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} and the latter the non-trivial semidirect product \mathbb{Z}/3\mathbb{Z} \rtimes_{\phi} \mathbb{Z}/2\mathbb{Z} where \phi is the inversion action.^[6]

Extension Problem

The extension problem in group theory asks: given groups N and Q, determine all groups E (up to isomorphism) such that there exists a short exact sequence $1 \to N \to E \to Q \to 1, where the inclusion N \hookrightarrow E has N normal in E and the projection E \twoheadrightarrow Q is surjective with kernel N.^[2]^[8] This problem seeks to construct and classify such intermediate groups E that "extend" Q by incorporating N as a normal subgroup. Two extensions $1 \to N \to E_1 \to Q \to 1 and $1 \to N \to E_2 \to Q \to 1 are equivalent if there exists an isomorphism \varphi: E_1 \to E_2 such that the diagram

\begin{CD} 1 @>>> N @>>> E_1 @>>> Q @>>> 1 \\ @. @| @V{\varphi}VV @| @. \\ 1 @>>> N @>>> E_2 @>>> Q @>>> 1 \end{CD}

commutes, meaning the maps compose equally on the rows.^[2]^[8] Equivalence thus preserves the structural relationship between N and Q, focusing on isomorphic realizations of the same extension data. For such an extension to exist, there must be a homomorphism \phi: [Q](/page/Q) \to \Aut(N) induced by conjugation in E: for q \in [Q](/page/Q) and n \in N, lifting q to some \tilde{q} \in E gives \phi(q)(n) = \tilde{q} n \tilde{q}^{-1}, defining an action of [Q](/page/Q) on N compatible with the group operations.^[8]^[3] In general, solutions to the extension problem are not unique up to isomorphism, as multiple non-equivalent E may arise for fixed N and Q; however, uniqueness holds in special cases, such as the trivial extension where E \cong N \times [Q](/page/Q).^[2]

Classification

Trivial Extensions

In group theory, a short exact sequence of groups $1 \to N \xrightarrow{i} E \xrightarrow{\pi} Q \to 1, where N is normal in E, is called a trivial extension if it splits (i.e., there exists a homomorphism s: Q \to E such that \pi \circ s = \mathrm{id}_Q) and the induced action of Q on N—given by conjugation in E via the identification of Q with its image under s—is trivial, meaning the homomorphism \phi: Q \to \Aut(N) is the zero map.^[9]^[10] Such extensions are characterized by the fact that E is isomorphic to the direct product N \times Q, where the group operation is componentwise: (n_1, q_1) \cdot (n_2, q_2) = (n_1 n_2, q_1 q_2), the inclusion i sends n \mapsto (n, 1_Q), and the projection \pi sends (n, q) \mapsto q.^[10] Trivial extensions form a subclass of split extensions, in which the conjugating action need not be trivial.^[9] A key condition for an extension to be trivial is that N lies in the center of E (ensuring the action is trivial) and the sequence admits a splitting homomorphism.^[10] In this case, elements of the image of s commute with every element of N. For example, any abelian group A arises as a trivial extension of itself by the trivial group \{1\}, via the sequence $1 \to A \xrightarrow{\mathrm{id}} A \to \{1\} \to 1, which splits canonically.^[10]

Split Extensions

In group theory, a split extension of a group Q by a normal subgroup N refers to a short exact sequence $1 \to N \xrightarrow{i} E \xrightarrow{\pi} Q \to 1 that admits a splitting homomorphism s: Q \to E satisfying \pi \circ s = \mathrm{id}_Q.^[11] This splitting implies that E contains a subgroup isomorphic to Q that intersects N trivially and generates E together with N.^[12] Such extensions are precisely those where E is isomorphic to the semidirect product N \rtimes_\phi Q, with \phi: Q \to \Aut(N) the homomorphism encoding the action of the image of s on N by conjugation.^[13] The semidirect product is constructed explicitly as the set N \times Q with the twisted multiplication

(n_1, q_1)(n_2, q_2) = \bigl( n_1 \cdot \phi(q_1)(n_2),\ q_1 q_2 \bigr),

where \cdot is the group operation in N.^[12] Trivial extensions arise as the special case where \phi is the trivial homomorphism, reducing to the direct product N \times Q.^[14] A representative example is the symmetric group S_3, which realizes the split extension \mathbb{Z}/3\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z} via the nontrivial action of \mathbb{Z}/2\mathbb{Z} on \mathbb{Z}/3\mathbb{Z} by inversion.^[14]

Non-Split Extensions

A non-split extension of a group Q by a normal subgroup N is a short exact sequence $1 \to N \to E \to Q \to 1 for which there does not exist a homomorphism s: Q \to E such that the composition with the projection E \to Q is the identity on Q.^[15] In such extensions, the middle group E cannot be isomorphic to a semidirect product N \rtimes Q, distinguishing non-split cases from those where a complementary subgroup isomorphic to Q exists within E.^[16] A concrete example is the quaternion group Q_8 of order 8, which realizes the non-split extension $1 \to \mathbb{Z}/2\mathbb{Z} \to Q_8 \to \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \to 1, with \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} denoting the Klein four-group; no such section s exists here.^[15] For contrast, the dihedral group D_4 of order 8 is a split extension of the same quotient by \mathbb{Z}/2\mathbb{Z}, arising as the semidirect product (\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}) \rtimes \mathbb{Z}/2\mathbb{Z}.^[17] The failure of an extension to split is obstructed by non-trivial 2-cocycles associated to the action of [Q](/page/Q) on [N](/page/N+), reflecting the non-existence of a compatible lifting homomorphism.^[18]

Central Extensions

Properties

A central extension of a group Q by a group N is a short exact sequence $1 \to N \xrightarrow{i} E \xrightarrow{\pi} Q \to 1 such that N is contained in the center Z(E) of E.^[18] This centrality condition implies that conjugation by elements of E fixes every element of N pointwise, i.e., for all e \in E and n \in N, e n e^{-1} = n.^[16] As a result, N must be abelian, since the commutator subgroup [N, N] is contained in [E, N] = \{1\}.^[16] The defining centrality of the extension ensures that the induced action \phi: Q \to \Aut(N), given by choosing a lift \tilde{q} \in E with \pi(\tilde{q}) = q and \phi(q)(n) = \tilde{q} n \tilde{q}^{-1} for q \in Q and n \in N, is trivial.^[19] These properties highlight the "central" nature of the kernel, distinguishing such extensions from more general ones where the action may be nontrivial. An equivalent characterization of central extensions is that they are precisely those short exact sequences where the image of the inclusion i: N \to E lands in the center Z(E).^[16] Regarding derived length, central extensions contribute to the structure of nilpotent groups: a group constructed via a finite sequence of central extensions admits a central series of corresponding length, and if this series reaches the whole group, the group is nilpotent of class equal to the series length.^[16] A representative example is the integer Heisenberg group H_3(\mathbb{Z}), consisting of $3 \times 3 upper-triangular matrices with integer entries and ones on the diagonal, which forms a central extension $1 \to \mathbb{Z} \to H_3(\mathbb{Z}) \to \mathbb{Z}^2 \to 1.^[20] Trivial extensions, such as direct products E = N \times Q, are central extensions featuring both trivial action and splitting.^[19]

Cohomology Classification

Group cohomology provides a powerful framework for classifying group extensions. Specifically, for central extensions of a group Q by an abelian group N (where Q acts trivially on N), the isomorphism classes of such extensions $1 \to N \to E \to Q \to 1 are in bijective correspondence with elements of the second cohomology group H^2(Q, N). Each element in H^2(Q, N) represents an equivalence class of extensions, where two extensions are equivalent if there is an isomorphism between them compatible with the projections to Q and inclusions from N.^[21] The classification arises from associating to each extension a 2-cocycle. Given a set-theoretic section s: Q \to E with s(1) = 1, the 2-cocycle f: Q \times Q \to N is defined by s(q_1) s(q_2) = \tilde{f}(q_1, q_2) s(q_1 q_2), where \tilde{f}(q_1, q_2) \in N. For the extension to be associative, f must satisfy the 2-cocycle condition:

f(q_1, q_2) \, f(q_1 q_2, q_3) = f(q_1, q_2 q_3) \, f(q_2, q_3)

for all q_1, q_2, q_3 \in Q, reflecting the trivial action on N. Two cocycles define equivalent extensions if they differ by a 2-coboundary, i.e., f(q_1, q_2) = \phi(q_1) \phi(q_2) \phi(q_1 q_2)^{-1} for some \phi: Q \to N, and the quotient Z^2(Q, N)/B^2(Q, N) = H^2(Q, N) parametrizes the classes.^[13] An extension splits, meaning E \cong N \rtimes Q (or direct product for central case), if and only if the corresponding cohomology class in H^2(Q, N) is zero, i.e., the cocycle is a coboundary. This criterion distinguishes split extensions from non-split ones, with non-trivial classes yielding inseparable structures.^[21] For general (non-central) extensions, where Q acts non-trivially on N via conjugation in E, the classification still uses H^2(Q, N), but now N is viewed as a Q-module under this action. The cocycle condition generalizes to account for the action:

f(q_1, q_2) \, f(q_1 q_2, q_3) = q_1 \cdot f(q_2, q_3) \, f(q_1, q_2 q_3),

ensuring compatibility with the module structure, and equivalence classes are again given by H^2(Q, N).^[13] A concrete example is the classification of central extensions of the cyclic group \mathbb{Z}/p\mathbb{Z} by \mathbb{Z}/p\mathbb{Z} for prime p, where H^2(\mathbb{Z}/p\mathbb{Z}, \mathbb{Z}/p\mathbb{Z}) \cong \mathbb{Z}/p\mathbb{Z}. The trivial class yields the direct product \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}, while non-trivial classes produce the cyclic group \mathbb{Z}/p^2\mathbb{Z}. For higher-dimensional cases, central extensions of the elementary abelian group (\mathbb{Z}/p\mathbb{Z})^2 by \mathbb{Z}/p\mathbb{Z} are classified by H^2((\mathbb{Z}/p\mathbb{Z})^2, \mathbb{Z}/p\mathbb{Z}) \cong (\mathbb{Z}/p\mathbb{Z})^3, with non-trivial classes yielding extraspecial p-groups of order p^3, such as the Heisenberg group over \mathbb{F}_p, characterized by a non-degenerate alternating bilinear form from the cocycle.^[22]

Applications

Lie Groups

In the context of Lie groups, an extension is defined as a short exact sequence $1 \to N \xrightarrow{i} G \xrightarrow{p} Q \to 1, where N, G, and Q are Lie groups, and the maps i and p are smooth Lie group homomorphisms such that i is injective, p is surjective, and the image of i equals the kernel of p.^[23] This structure preserves the differentiable manifold aspects of the groups involved, ensuring that G inherits a smooth group operation compatible with the topology and the extension maps. Such extensions are fundamental in Lie theory for constructing larger groups from smaller ones while maintaining the smooth homomorphism properties essential for infinitesimal analysis via Lie algebras. Central extensions play a prominent role in Lie theory, particularly through the concept of universal central extensions, which provide canonical ways to enlarge a Lie algebra by a central ideal in a manner that captures all possible central extensions. For instance, affine Kac-Moody algebras arise as the universal central extensions of loop algebras associated to finite-dimensional simple Lie algebras, where the loop algebra \mathfrak{g}((t)) (functions from the circle to \mathfrak{g}) is extended centrally by a one-dimensional ideal generated by a canonical central element.^[24] This construction is crucial in infinite-dimensional Lie theory and has applications in conformal field theory and string theory. A concrete finite-dimensional example is the double cover \mathbb{Z}/2\mathbb{Z} \to \mathrm{SU}(2) \to \mathrm{SO}(3), where \mathrm{SU}(2) serves as the universal cover of the rotation group \mathrm{SO}(3), with the kernel acting centrally; this extension resolves topological obstructions in representations of rotations, such as in quantum mechanics for spin-1/2 particles.^[25] While an extension of Lie algebras may split (admitting a Lie algebra homomorphism as a section), the corresponding Lie group extension does not always split smoothly due to topological constraints, even if it splits abstractly as groups. For connected Lie groups, continuous splitting fails when the extension class lies in a non-trivial element of the second continuous cohomology group H^2(Q; N), reflecting mismatches between algebraic and topological structures; examples include certain extensions of compact Lie groups where the fundamental group introduces non-trivial topology.^[26] In general, any short exact sequence of Lie groups induces a short exact sequence of their associated Lie algebras, \mathfrak{n} \to \mathfrak{g} \to \mathfrak{q}, via differentiation of the smooth maps at the identity, allowing the infinitesimal structure of the extension to be analyzed independently of global topological issues.^[23]

Topological Groups

In the context of topological groups, a group extension is termed topological if the groups A, E, and G are equipped with compatible topologies making them topological groups, and the inclusion i: A \to E and projection \pi: E \to G are continuous homomorphisms. This setup ensures that the extension respects both algebraic and topological structures, allowing for the study of properties like local compactness, connectedness, and the behavior of homomorphisms under continuity constraints. Such extensions arise naturally in settings where groups carry additional geometric or analytic structure, such as Lie groups or spaces in algebraic topology.^[27] A key distinction in topological extensions is the notion of continuous splitting, which requires the existence of a continuous group homomorphism s: G \to E serving as a section, i.e., \pi \circ s = \mathrm{id}_G. This is stricter than an algebraic splitting, where a mere set-theoretic or algebraic section exists without continuity. Continuous splittings may fail even when algebraic ones exist, due to topological obstructions like the density of subgroups or incompatibility with compactness. For example, consider the extension $1 \to \mathbb{Q}/\mathbb{Z} \to E \to \mathbb{R}/\mathbb{Z} \to 1, where E = \mathbb{R} \times (\mathbb{Q}/\mathbb{Z}) with the product topology, and the projection \pi(x, y) = x + \mathbb{Z} + y (identifying \mathbb{Q}/\mathbb{Z} with a dense subgroup of \mathbb{R}/\mathbb{Z}). Algebraically, this splits via a choice of representatives for cosets in \mathbb{Q}, but no continuous splitting exists because any continuous section would map the compact group \mathbb{R}/\mathbb{Z} to a compact subgroup of E, which contradicts the density of \mathbb{Q}/\mathbb{Z} in the torus.^[26] A prominent example of a topological central extension with continuous issues is the universal covering sequence $1 \to \mathbb{Z} \to \mathbb{R} \to S^1 \to 1, where \mathbb{Z} carries the discrete topology, \mathbb{R} and S^1 (the circle group) have their standard topologies, the inclusion sends n \mapsto n, and the projection is the exponential map t \mapsto e^{2\pi i t}. This is central since [\mathbb{R}, \mathbb{R}] = 0 and \mathbb{Z} lies in the center. Although local continuous sections exist (reflecting the local triviality of the covering), there is no global continuous section, as it would imply a continuous embedding of the compact S^1 into the contractible \mathbb{R} as a subgroup complementing \mathbb{Z}, which is impossible due to topological invariants like non-trivial homotopy. This extension highlights obstructions in covering spaces, where the failure of global continuity relates to the non-trivial fundamental group of S^1. Note that smooth versions of such extensions appear in Lie groups, but the topological framework applies more broadly. The classification of topological group extensions, particularly central ones, relies on continuous cohomology. The second continuous cohomology group H^2_{\mathrm{cont}}(G, A) classifies equivalence classes of topological extensions of G by A up to topological equivalence, where cocycles are continuous maps satisfying the cocycle condition \omega(gh, a) = \omega(g, h \cdot a) \cdot g \cdot \omega(h, a) (for left actions), and coboundaries arise from continuous 1-cochains. For locally compact groups, this cohomology is computed using continuous cochains on compact supports or inductive limits. In broader settings, such as paracompact spaces, sheaf cohomology of the constant sheaf \underline{A} on the classifying space BG provides an alternative classification, capturing topological obstructions via higher derived functors.^[27]^[28] Applications of topological group extensions appear prominently in algebraic topology, particularly in the study of covering spaces and their relation to fundamental groups. For a path-connected, locally path-connected covering space p: \tilde{X} \to X, the sequence $1 \to \pi_1(\tilde{X}, \tilde{x_0}) \to \pi_1(X, x_0) \to \mathrm{Aut}(p) \to 1 forms a (possibly split) extension of the deck transformation group by the fundamental group of the cover, where the action is continuous when topologies are considered. This framework classifies connected coverings via subgroups of \pi_1(X), with non-split extensions corresponding to non-trivial monodromy.

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