Let G be a group. If B and A are G-modules, make A ⊗ℤ B into a G-module with diagonal action: g(

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Let G be a group. If B and A are G-modules, make A ⊗ℤ B into...

Let $G$ be a group. If $B$ and $A$ are $G$-modules, make $A \otimes_{\mathbb{Z}} B$ into a $G$-module with diagonal action: $$ g(b \otimes a)=(g b) \otimes(g a) . $$ If $A$ is a $G$-module, let $A_{0}$ be its underlying abelian group. Prove that $\mathbb{Z} G \otimes_{\mathbb{Z}} A_{0} \cong \mathbb{Z} G \otimes_{\mathbb{Z}} A$ as $G$-modules. Hint. Define $f: \mathbb{Z} G \otimes_{\mathbb{Z}} A_{0} \rightarrow \mathbb{Z} G \otimes_{\mathbb{Z}} A$ by $g \otimes a \mapsto g \otimes g a$.

Let $G$ be a group. If $B$ and $A$ are $G$-modules, make $A \otimes_{\mathbb{Z}} B$ into a $G$-module with diagonal action:
$$
g(b \otimes a)=(g b) \otimes(g a) .
$$
If $A$ is a $G$-module, let $A_{0}$ be its underlying abelian group. Prove that $\mathbb{Z} G \otimes_{\mathbb{Z}} A_{0} \cong \mathbb{Z} G \otimes_{\mathbb{Z}} A$ as $G$-modules.
Hint. Define $f: \mathbb{Z} G \otimes_{\mathbb{Z}} A_{0} \rightarrow \mathbb{Z} G \otimes_{\mathbb{Z}} A$ by $g \otimes a \mapsto g \otimes g a$.

Key Concepts

Underlying Abelian Group

The underlying abelian group of a G-module is the structure that remains when the additional group action is ignored. It is essentially the module considered solely as an abelian group without any reference to the extra structure provided by the group action. This concept allows one to compare or manipulate the module's algebraic structure independent of the group action.

Diagonal Action

The diagonal action is a natural way to define a group action on the tensor product of two G-modules. For an element g in G and a tensor product element a ? b, the group acts by g(a ? b) = (g·a) ? (g·b). This allows one to extend the group action defined on individual modules to their tensor product in a manner that respects the module structures.

Module Isomorphism

A module isomorphism is a bijective homomorphism between modules that preserves the module operations, including scalar or group actions. When two modules are isomorphic, they are structurally the same in the context of the module theory, meaning that many properties and problems concerning one can be translated and solved in the other.

G-module

A G-module is an abelian group that is equipped with an action by a group G. This action is compatible with the abelian group structure, meaning that for each element of G, the map induced on the group is an automorphism. G-modules generalize the idea of representations of groups, allowing one to study group actions on abelian groups or vector spaces.

Group

A group is an algebraic structure consisting of a set equipped with an operation that combines any two elements to form a third element, satisfying the four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverse elements. Groups serve as the basic building blocks in abstract algebra and underpin many structures including modules and group actions.

Tensor Product

The tensor product of two modules over a ring (here, abelian groups regarded as modules over the ring of integers) is a construction that produces a new module. It encodes bilinear maps between the original modules and allows the study of how modules interact. In the context of G-modules, the tensor product can be given additional structure by defining a G-action that interacts with the factors.

Mapping Function for Isomorphisms

The mapping function in the context of module isomorphisms is a specific homomorphism designed to show the equivalence of two module structures. In this approach, the function defined by sending g ? a to g ? g·a demonstrates how the module structures interact and can be transformed into one another, which is critical in establishing an isomorphism between the two tensor products with differing perspectives on the action.

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