G -module
In abstract algebra, a G-module (or G-module over the integers) is an abelian group M equipped with a left action of a group G on M by group endomorphisms, meaning there is a homomorphism G → Aut(M) where Aut(M) denotes the automorphism group of M as an abelian group.[1] Equivalently, a G-module is a module over the integral group ring ℤ[G], where elements of ℤ[G] act on M via the ring's multiplication extended from the G-action.[2] This structure captures compatible group actions on additive groups and forms the foundation for studying symmetries in algebraic settings.[3] In the context of representation theory, the notion generalizes to R-G-modules for a commutative ring R, where M is an R-module and G acts by R-linear endomorphisms; when R is a field k, this yields a k[G]-module, or vector space over k with a linear G-action, often simply called a representation of G over k.[4] Key examples include the regular representation, where M = ℤ[G] with left multiplication by group elements, and the trivial representation, where g · m = m for all g ∈ G and m ∈ M.[5] Subrepresentations correspond to G-invariant subgroups or subspaces, and irreducible G-modules—those with no nontrivial submodules—are fundamental for decomposing general modules into direct sums via Maschke's theorem when G is finite and k has characteristic not dividing |G|.[4] G-modules play a central role in group cohomology and homology, where the cohomology groups Hn(G, M) measure extensions and obstructions related to the G-action on M, with applications in algebraic topology, number theory, and the study of group extensions.[1] Characters, defined as the trace of the action maps, provide invariants for classifying representations up to isomorphism, particularly for finite groups.[4] The category of G-modules admits tensor products, direct sums, and Hom functors that respect the action, enabling the development of homological algebra tailored to group symmetries.[5]Fundamentals
Definition
A G-module, also known as a module over the group G, is an abelian group M equipped with a left action \rho: G \times M \to M that is compatible with the abelian group structure of M [12]. Specifically, for all g, h \in G and m, n \in M, the action satisfies \rho(g, m + n) = \rho(g, m) + \rho(g, n) and \rho(g, \rho(h, m)) = \rho(gh, m), with the identity element of G acting as the identity map on M [12]. This action turns M into a module in the sense that elements of G act as group automorphisms on M [2]. Equivalently, the action defines a group homomorphism \rho: G \to \Aut(M), where \Aut(M) denotes the automorphism group of the abelian group M . This perspective emphasizes that the group action is a representation of G by automorphisms preserving the additive structure of M . Such G-modules are in one-to-one correspondence with modules over the group ring \mathbb{Z}G, providing an algebraic framework for studying group actions . A morphism of G-modules between two G-modules M and N is a group homomorphism f: M \to N that is G-equivariant, meaning f(\rho(g, m)) = \rho(g, f(m)) for all g \in G and m \in M . These morphisms preserve the group action and form the arrows in the category of G-modules, often denoted \Mod_G or _G\Mod . The category of G-modules is an abelian category, where kernels, cokernels, and exact sequences are defined in the standard way for module categories . This structure allows for the application of homological algebra techniques to G-modules, treating them as objects in a rich categorical framework .Relation to Group Rings
The group ring \mathbb{Z}G consists of all formal sums \sum_{g \in G} a_g g, where a_g \in \mathbb{Z} and only finitely many a_g are nonzero, equipped with addition componentwise and multiplication defined by extending the group operation linearly: \left( \sum a_g g \right) \left( \sum b_h h \right) = \sum_{g,h \in G} a_g b_h (gh).[6] This structure makes \mathbb{Z}G a ring with unity $1 = e, where e is the identity element of G.[7] Every G-module M (an abelian group with a compatible G-action) corresponds bijectively to a left \mathbb{Z}G-module structure on M, given by the action \left( \sum_{g \in G} a_g g \right) \cdot m = \sum_{g \in G} a_g (g \cdot m) for all m \in M.[8] This correspondence is an equivalence of categories between the category of G-modules and the category of left \mathbb{Z}G-modules.[6] This equivalence is functorial in nature. The forgetful functor from the category of left \mathbb{Z}G-modules to the category of abelian groups, which forgets the \mathbb{Z}G-action, admits a left adjoint given by induction: for an abelian group A, the induced module is \mathbb{Z}G \otimes_{\mathbb{Z}} A, and a right adjoint given by coinduction: \mathrm{Hom}_{\mathbb{Z}}(\mathbb{Z}G, A). For a general commutative ring R with unity, the construction generalizes by replacing \mathbb{Z} with R: the group ring RG consists of formal R-linear combinations \sum_{g \in G} a_g g with finite support, and G-modules over R (i.e., R-modules with compatible G-action) are equivalent to left RG-modules via the analogous action formula.[6]Properties
Submodules and Quotients
In the category of G-modules, a submodule N of a G-module M is an abelian subgroup of M that is closed under the G-action, meaning g \cdot n \in N for all g \in G and n \in N.[9] This ensures that N itself inherits a well-defined G-module structure from M.[10] Given a submodule N \subseteq M, the quotient M/N is defined as the set of cosets \{m + N \mid m \in M\} with the induced abelian group operation. The G-action on M/N is given by g \cdot (m + N) = (g \cdot m) + N for g \in G and m \in M, which is well-defined precisely because N is G-invariant.[9] This makes M/N a [G](/page/G)-module in a natural way.[11] A sequence of G-modules \cdots \to M_{i-1} \xrightarrow{f_{i-1}} M_i \xrightarrow{f_i} M_{i+1} \to \cdots with G-equivariant homomorphisms f_i: M_i \to M_{i+1} (i.e., f_i(g \cdot m) = g \cdot f_i(m) for all g \in G, m \in M_i) is exact if the underlying sequence of abelian groups is exact, meaning \operatorname{im} f_{i-1} = \ker f_i for each i.[9] In particular, a short exact sequence $0 \to N \to M \to Q \to 0consists ofG-equivariant maps where the first is injective, the second surjective, and \operatorname{im}(N \to M) = \ker(M \to Q)$.[12] Direct sums and products of G-modules are formed componentwise with respect to the diagonal G-action: for G-modules M_i (i \in I), the direct sum \bigoplus_{i \in I} M_i has elements finite sums \sum m_i with m_i \in M_i and g \cdot (\sum m_i) = \sum (g \cdot m_i), while the direct product \prod_{i \in I} M_i has elements (m_i)_{i \in I} with g \cdot (m_i) = (g \cdot m_i)_{i \in I}.[12] These constructions preserve the G-module structure exactly as in the underlying category of abelian groups.[9]Morphisms and Categories
The category of G-modules, denoted \mathrm{Mod}_G or G-\mathrm{Mod}, consists of all abelian groups equipped with a left action of the group G as objects, with morphisms given by G-equivariant group homomorphisms, i.e., additive maps f: M \to N satisfying f(g \cdot m) = g \cdot f(m) for all g \in G and m \in M. This category is equivalent to the category of left modules over the group ring \mathbb{Z}G, and as such, it inherits the structure of an abelian category: every monomorphism is the kernel of its cokernel, every epimorphism is the cokernel of its kernel, and kernels and cokernels exist and coincide with their images. In \mathrm{Mod}_G, the kernel of a morphism f: M \to N is the usual kernel \{m \in M \mid f(m) = 0\} as an abelian group, which is automatically a G-submodule since equivariance implies g \cdot \ker f \subseteq \ker f; similarly, cokernels are quotients by G-submodule images.[13] The forgetful functor F: \mathrm{Mod}_G \to \mathrm{Ab}, which sends a G-module to its underlying abelian group and a G-equivariant map to the corresponding group homomorphism, is exact: a sequence of G-modules is exact if and only if the underlying sequence of abelian groups is exact, as the G-action does not affect the computation of kernels and cokernels in the underlying category. The Hom-sets \mathrm{Hom}_G(M, N) are the abelian groups of all G-equivariant homomorphisms from M to N, and for fixed M, the covariant functor \mathrm{Hom}_G(M, -): \mathrm{Mod}_G \to \mathrm{Ab} is left exact, preserving finite limits such as kernels of exact sequences $0 \to N' \to N \to N'', yielding $0 \to \mathrm{Hom}_G(M, N') \to \mathrm{Hom}_G(M, N) \to \mathrm{Hom}_G(M, N''). Dually, \mathrm{Hom}_G(-, N) is contravariant and left exact.[13] The category \mathrm{Mod}_G has enough projective objects, meaning that for every G-module M, there exists a surjection P \twoheadrightarrow M from a projective G-module P; the free \mathbb{Z}G-modules \mathbb{Z}G \otimes_{\mathbb{Z}} A \cong \bigoplus_A \mathbb{Z}G for abelian groups A are projective, and every projective G-module is a direct summand of a free one, allowing projective resolutions for computing derived functors. Similarly, \mathrm{Mod}_G has enough injective objects, so every G-module embeds into an injective one, facilitating injective resolutions; injective G-modules can be constructed as direct sums or products of indecomposable injectives over \mathbb{Z}G. These properties ensure that \mathrm{Mod}_G supports the full machinery of homological algebra, including Ext and Tor functors.[13]Examples and Constructions
Trivial and Regular Modules
The trivial G-module is defined as any abelian group M equipped with a G-action such that g \cdot m = m for all g \in G and m \in M.[1] This action renders every element of G acting as the identity map on M, making it the simplest possible G-module structure.[14] A standard example is the integers \mathbb{Z} viewed as a trivial G-module, where the action ignores group elements entirely.[1] The regular G-module is the group ring \mathbb{Z}G endowed with the left multiplication action, where g \cdot \left( \sum_{h \in G} n_h h \right) = \sum_{h \in G} n_h (g h) for g \in G and coefficients n_h \in \mathbb{Z}.[14] This module captures the full structure of G acting on its formal linear combinations. For a finite group G, when considered over the complex numbers \mathbb{C}, the regular module \mathbb{C}G decomposes as a direct sum \bigoplus_{\rho} (\dim \rho) \cdot \rho, where the sum runs over all irreducible representations \rho of G.[14] For a finite group G and subgroup H \leq G, the permutation module on the cosets G/H is the free abelian group \mathbb{Z}[G/H] generated by the left cosets, with G acting by permutation: g \cdot (kH) = (g k) H for g \in G and coset representative k \in G.[1] This yields a G-module isomorphic to the induced module from the trivial H-module \mathbb{Z}, providing a concrete permutation representation.[1] In the context of vector spaces, an example arises with the orthogonal group O(n) acting on \mathbb{R}^n via matrix multiplication, where g \cdot v = g v for g \in O(n) and v \in \mathbb{R}^n.[15] This defines a O(n)-module structure that preserves the standard Euclidean inner product, illustrating a faithful linear action.[15]Induced and Coinduced Modules
In representation theory of groups, given a group G and a subgroup H \leq G, important constructions allow one to build G-modules from H-modules. The induced module provides a way to extend an H-module to a G-module via tensor product over the group ring.[16] Specifically, for an H-module N, the induced module is defined as \operatorname{Ind}_H^G N = \mathbb{Z}G \otimes_{\mathbb{Z}H} N, where the G-action is given by g \cdot (x \otimes n) = gx \otimes n for g, x \in G and n \in N. This construction leverages the bimodule structure of \mathbb{Z}G over \mathbb{Z}G and \mathbb{Z}H.[17] Dually, the coinduced module extends H-modules using Hom spaces. For an H-module N, the coinduced module is \operatorname{Coind}_H^G N = \operatorname{Hom}_{\mathbb{Z}H}(\mathbb{Z}G, N), equipped with the G-action (g \cdot f)(x) = f(x g^{-1}) for g, x \in G and f \in \operatorname{Hom}_{\mathbb{Z}H}(\mathbb{Z}G, N). This action ensures compatibility with the H-module structure on N, reflecting the right G-module structure on \mathbb{Z}G.[16] These functors participate in adjunctions with the restriction functor \operatorname{Res}_H^G, which forgets the [G](/page/G)-action to yield an [H](/page/H+)-module. The induction functor is left adjoint to restriction: \operatorname{Hom}_G(\operatorname{Ind}_H^G N, M) \cong \operatorname{Hom}_H(N, \operatorname{Res}_H^G M) for any [G](/page/G)-module M and [H](/page/H+)-module N. Symmetrically, coinduction is right adjoint to restriction: \operatorname{Hom}_G(M, \operatorname{Coind}_H^G N) \cong \operatorname{Hom}_H(\operatorname{Res}_H^G M, N). These isomorphisms, known as Frobenius adjunctions or Shapiro's lemma in certain contexts, underpin many applications in representation theory.[17] When the index [G : H] is finite, the induced and coinduced modules are naturally isomorphic as G-modules. This isomorphism arises from the finite sum over coset representatives and preserves the module structures, facilitating computations in both directions.[16]Structure and Homological Algebra
Invariants and Coinvariants
In a G-module M, the submodule of invariants, denoted M^G, consists of those elements fixed by the entire group action: M^G = \{ m \in M \mid g \cdot m = m \ \forall \, g \in G \}. This forms a submodule of M on which G acts trivially via the identity map.[12] The module of coinvariants, denoted M_G, is the quotient of M by the submodule generated by all elements of the form g \cdot m - m for g \in G and m \in M: M_G = M / \langle g \cdot m - m \mid g \in G, \, m \in M \rangle. This construction yields the largest quotient module of M on which G acts trivially, effectively identifying elements within each G-orbit.[12] When G is finite, a key connection between invariants and coinvariants arises via the norm map N: [M^G](/page/M&G) \to M_G, defined by N(m) = \sum_{g \in [G](/page/G)} g \cdot m modulo the relations generating the coinvariants submodule (which simplifies to |[G](/page/G)| \cdot m in the quotient since m is fixed). This map plays a central role in explicit computations involving these constructions, such as in group homology and the study of fixed-point subspaces.Resolutions and Ext Functors
In homological algebra, a projective resolution of a left \mathbb{Z}[G](/page/G)-module M is an exact sequence \cdots \to P_1 \to P_0 \to M \to 0, where each P_i is a projective \mathbb{Z}[G](/page/G)-module, typically free since free modules over group rings are projective.[13] These resolutions provide a framework for computing derived functors associated to M, enabling the study of extensions and homology in the category of G-modules. Projective \mathbb{Z}[G](/page/G)-modules, such as free modules of rank n generated by basis elements with diagonal G-action, serve as building blocks for such resolutions.[18] The Ext functors \operatorname{Ext}^n_{\mathbb{Z}G}(M, N) for G-modules M and N are the right derived functors of the Hom functor \operatorname{Hom}_{\mathbb{Z}G}(-, -), measuring the n-fold extensions of M by N. They are computed by applying \operatorname{Hom}_{\mathbb{Z}G}(-, N) to a projective resolution P_\bullet \to M of M, yielding \operatorname{Ext}^n_{\mathbb{Z}G}(M, N) \cong H^n \bigl( \operatorname{Hom}_{\mathbb{Z}G}(P_\bullet, N) \bigr), where the cohomology is taken after deleting the identity map P_0 \to M. This construction captures obstructions to lifting homomorphisms and is central to understanding module extensions in group representations.[13][18] Dually, the Tor functors \operatorname{Tor}_n^{\mathbb{Z}G}(M, N) are the left derived functors of the tensor product M \otimes_{\mathbb{Z}G} N, relating to the homology of tensor products over the group ring. Given a projective resolution P_\bullet \to M of M, they are given by \operatorname{Tor}_n^{\mathbb{Z}G}(M, N) \cong H_n \bigl( P_\bullet \otimes_{\mathbb{Z}G} N \bigr), with homology computed after deleting the augmentation P_0 \to M. These functors quantify the failure of exactness in tensor products and arise naturally in the study of bilinear forms invariant under G-actions.[13][18] For a finite group G, the standard bar resolution provides an explicit projective resolution of the trivial \mathbb{Z}G-module \mathbb{Z}, where G acts trivially. It is the complex \cdots \to P_1 \to P_0 \to \mathbb{Z} \to 0 with P_n the free \mathbb{Z}G-module on basis elements [g_0 | \cdots | g_n] for g_i \in G, and differential d_n [g_0 | \cdots | g_n] = \sum_{i=0}^n (-1)^i [g_0 | \cdots | \hat{g_i} | \cdots | g_n] + \sum_{i=0}^{n-1} (-1)^i [g_0 | \cdots | g_i g_{i+1} | \cdots | g_n], where the hat denotes omission; this resolution is exact and functorial, facilitating computations of Ext and Tor groups involving the trivial module.[19][18]Applications
Group Cohomology
Group cohomology is a fundamental tool in homological algebra that studies the structure of groups through their actions on abelian groups, utilizing G-modules as coefficients. For a discrete group G and a left ℤG-module M, the nth cohomology group is defined asH^n(G, M) = \Ext^n_{\mathbb{Z}G}(\mathbb{Z}, M),
where ℤ denotes the trivial ℤG-module with G acting via the identity map. This definition arises from applying the right derived functors of the Hom functor to a projective resolution of the trivial module ℤ over the group ring ℤG.[20] In low dimensions, these groups admit concrete interpretations. The zeroth cohomology group is the subgroup of invariants,
H^0(G, M) = M^G = \{ m \in M \mid g \cdot m = m \ \forall g \in G \},
consisting of elements fixed by the entire group action. The first cohomology group classifies crossed homomorphisms from G to M modulo principal ones: a crossed homomorphism f: G → M satisfies f(gh) = f(g) + g · f(h) for all g, h ∈ G, and principal crossed homomorphisms are those of the form g ↦ g · m - m for some fixed m ∈ M. Thus,
H^1(G, M) = Z^1(G, M) / B^1(G, M),
where Z^1 and B^1 denote the groups of crossed and principal crossed homomorphisms, respectively. This group relates to conjugacy classes of group extensions, as elements of H^1(G, M) parametrize automorphisms of extensions classified by H^2(G, M), up to inner automorphisms induced by the module.[20][12][21] A key technical tool is Shapiro's lemma, which relates cohomology of subgroups to that of the full group via coinduced modules. For a subgroup H ≤ G and an ℤH-module N, the coinduced module is
\Coind_H^G N = \Hom_{\mathbb{Z}H}(\mathbb{Z}G, N),
with G-action given by (g · f)(x) = f(x g) for f ∈ \Coind_H^G N and x ∈ ℤG. Shapiro's lemma states that
H^*(G, \Coind_H^G N) \cong H^*(H, N)
naturally in both G and N, providing a powerful method to compute cohomology by reducing to subgroups.[20]