Math 8246 Homework 4 PJW Date due: Monday March 26, 2007
... within J gives L the structure of a ZG-module. Suppose that M is another ZGmodule and that θ : L → M is a group homomorphism. Form the semidirect product M ⋊ J and let U = {(−θ(x), x) x ∈ L} ⊆ M ⋊ J. (Here M is written additively and J acts on M via the homomorphism J → G). (i) Show that U is a n ...
... within J gives L the structure of a ZG-module. Suppose that M is another ZGmodule and that θ : L → M is a group homomorphism. Form the semidirect product M ⋊ J and let U = {(−θ(x), x) x ∈ L} ⊆ M ⋊ J. (Here M is written additively and J acts on M via the homomorphism J → G). (i) Show that U is a n ...
2.1 Modules and Module Homomorphisms
... A-modules correspond to ring homomorphisms from A into endomorphism rings of abelian groups. Examples: (1) If I A then I becomes an A-module by regarding the ring multiplication, of elements of A with elements of I , as scalar multiplication. ...
... A-modules correspond to ring homomorphisms from A into endomorphism rings of abelian groups. Examples: (1) If I A then I becomes an A-module by regarding the ring multiplication, of elements of A with elements of I , as scalar multiplication. ...
Math 670 HW #2
... (b) Prove the universal property of the exterior product (feel free to assume the universal property of the tensor product). 2. Let A : V → W be a linear map between vector spaces. (a) Show that the induced map Λk (V ) → Λk (W ) is well-defined by v1 ∧ . . . ∧ vk 7→ Av1 ∧ . . . ∧ Avk (extending line ...
... (b) Prove the universal property of the exterior product (feel free to assume the universal property of the tensor product). 2. Let A : V → W be a linear map between vector spaces. (a) Show that the induced map Λk (V ) → Λk (W ) is well-defined by v1 ∧ . . . ∧ vk 7→ Av1 ∧ . . . ∧ Avk (extending line ...
Algebra Autumn 2013 Frank Sottile 24 October 2013 Eighth Homework
... (Here, [m] := {1, . . . , m} and the same for [n]. Show that this defines an action of Sm ≀ Sn on [m] × [n]. (b) Using this action or any other methods show that S2 ≀ S2 ≃ D8 , the dihedral group with 8 elements. (c) This action realizes S3 ≀ S2 as a sugroup of S6 . What are the cycle types of permu ...
... (Here, [m] := {1, . . . , m} and the same for [n]. Show that this defines an action of Sm ≀ Sn on [m] × [n]. (b) Using this action or any other methods show that S2 ≀ S2 ≃ D8 , the dihedral group with 8 elements. (c) This action realizes S3 ≀ S2 as a sugroup of S6 . What are the cycle types of permu ...
Lecture 1. Modules
... and R-action is given by r(m + N ) = rm + N for all r ∈ R, m ∈ M. Definition. If M and N are R-modules, a mapping ϕ : M → N is called a homomorphism of R-modules (alternatively ϕ is an R-linear mapping) if (1) ϕ is a homomorphism of abelian group (2) ϕ(rm) = rϕ(m) for all r ∈ R, m ∈ M . 1.4. Modules ...
... and R-action is given by r(m + N ) = rm + N for all r ∈ R, m ∈ M. Definition. If M and N are R-modules, a mapping ϕ : M → N is called a homomorphism of R-modules (alternatively ϕ is an R-linear mapping) if (1) ϕ is a homomorphism of abelian group (2) ϕ(rm) = rϕ(m) for all r ∈ R, m ∈ M . 1.4. Modules ...
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.