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G-module

Algebraic structure

In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G. Group (co)homology provides an important set of tools for studying general G-modules.

The term G-module is also used for the more general notion of an R-module on which G acts linearly (i.e. as a group of R-module automorphisms).

Definition and basics

Let G be a group. A left G-module consists of an abelian group M together with a left group action \rho:G\times M\to M such that :g\cdot(a_1+a_2)=g\cdot a_1+g\cdot a_2 for all a_1 and a_2 in M and all g in G, where g\cdot a denotes \rho(g,a). A right G-module is defined similarly. Given a left G-module M, it can be turned into a right G-module by defining a\cdot g=g^{-1}\cdot a.

A function f:M\rightarrow N is called a morphism of G-modules (or a G-linear map, or a G-homomorphism) if f is both a group homomorphism and G-equivariant.

The collection of left (respectively right) G-modules and their morphisms form an abelian category G\textbf{-Mod} (resp. \textbf{Mod-}G). The category G\text{-Mod} (resp. \text{Mod-}G) can be identified with the category of left (resp. right) \mathbb{Z}G-modules, i.e. with the modules over the group ring \mathbb{Z}[G].

A submodule of a G-module M is a subgroup A\subseteq M that is stable under the action of G, i.e. g\cdot a\in A for all g\in G and a\in A. Given a submodule A of M, the quotient module M/A is the quotient group with action g\cdot (m+A)=g\cdot m+A.

Examples

Given a group G, the abelian group \mathbb{Z} is a G-module with the trivial action g\cdot a=a.
Let M be the set of binary quadratic forms f(x,y)=ax^2+2bxy+cy^2 with a,b,c integers, and let G=\text{SL}(2,\mathbb{Z}) (the 2×2 special linear group over \mathbb{Z}). Define ::(g\cdot f)(x,y)=f((x,y)g^t)=f\left((x,y)\cdot\begin{bmatrix} \alpha & \gamma \ \beta & \delta \end{bmatrix}\right)=f(\alpha x+\beta y,\gamma x+\delta y), :where

\alpha & \beta \\ \gamma & \delta \end{bmatrix} :and (x,y)g is matrix multiplication. Then M is a G-module studied by Gauss. Indeed, we have :: g(h(f(x,y))) = gf((x,y)h^t)=f((x,y)h^tg^t)=f((x,y)(gh)^t)=(gh)f(x,y). - If V is a representation of G over a field K, then V is a G-module (it is an abelian group under addition). ## Topological groups If G is a topological group and M is an abelian topological group, then a **topological *G*-module** is a G-module where the action map G\times M\rightarrow M is continuous (where the product topology is taken on G\times M). In other words, a topological G-module is an abelian topological group M together with a continuous map G\times M\rightarrow M satisfying the usual relations g(a+a')=ga+ga', (gg')a=g(g'a), and 1a=a. ## Notes ## References - Chapter 6 of ## References 1. (1988). ["Representation Theory of Finite Groups and Associative Algebras"](https://archive.org/details/representationth11curt). *John Wiley & Sons*. 2. (1999). "Integral Quadratic Forms and Lattices: Proceedings of the International Conference on Integral Quadratic Forms and Lattices, June 15–19, 1998, Seoul National University, Korea". *American Mathematical Soc.*. 3. D. Wigner. (1973). "Algebraic cohomology of topological groups". *Trans. Amer. Math. Soc.*. ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/G-module) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/G-module?action=history). ::

group-theory representation-theory-of-groups

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