Help with Plane Lattices Problems in Abstract Algebra

In summary: A^-1(x-b). Since A is an element of the holohedry of L, we know that A^-1 is also a lattice map. Therefore, (A,b)(y) = A(A^-1(x-b)) + b = x. This shows that (A,b)(L) = L, and thus, (A,b) is a symmetry of L.3. In summary, the set of squares in Sn forms a subgroup for n = 2, 3, 4, 5. This can be proven by showing that the set of squares satisfies the four subgroup axioms: closure, associativity, identity, and inverse. Closure can be shown by proving that the product of two squares is also a
  • #1
Proggy99
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I have been struggling through this Abstract Algebra class and have completely bogged down in the Wallpaper Patterns chapter, especially the plane lattices section. Can anyone give me some help for the following three problems? I am not sure how to start any of the three problems. Thanks for any assistance!

1. Let L be a lattice in R[tex]^n[/tex] and g and <I think “and” should be “an” here?> element of E(n). Show that g maps L onto L if and only if gx and g[tex]^-^1[/tex]x are in L for each x in L.


2. Show that (A,b) is a symmetry of the lattice L if and only if b is in L, and A is in the holohedry of L.


3.Show that the set of squares in Sn is a subgroup for n = 2,3,4,5.
 
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  • #2


Sure, I'd be happy to offer some assistance with these problems!

1. To show that g maps L onto L, we need to prove that for every element y in L, there exists an element x in L such that g(x) = y. In other words, g is a surjective map from L to itself.

First, assume that g maps L onto L. Then for any y in L, there exists an x in L such that g(x) = y. This means that g^-1(y) = x. Since g^-1 is also a lattice map, we know that g^-1(y) is in L. So we have shown that g^-1x is in L for every x in L.

Conversely, assume that g^-1x is in L for every x in L. Now, let y be any element in L. Since g^-1x is in L for every x in L, we know that g^-1y is also in L. This means that g^-1y is an element of L that maps to y under g. Therefore, g is a surjective map from L to itself, and thus, g maps L onto L.

2. To show that (A,b) is a symmetry of L, we need to prove that (A,b) is a lattice map and that (A,b)(L) = L. In other words, (A,b) preserves the structure of the lattice.

First, let's show that (A,b) is a lattice map. This means that (A,b) preserves the lattice operations, namely addition and scalar multiplication. Let x and y be elements in L. Then (A,b)(x+y) = Ax + b + Ay + b = A(x+y) + 2b. Since L is closed under addition, x+y is also in L. Similarly, (A,b)(kx) = Akx + b = k(Ax) + kb. Since L is closed under scalar multiplication, kx is also in L. Therefore, (A,b) is a lattice map.

Now, let's show that (A,b)(L) = L. This means that for every x in L, there exists a y in L such that (A,b)(y) = x. Let x be any element in L. Since b is in L, we know that x-b is also in L. Now, let y =
 

1. What is a plane lattice in abstract algebra?

A plane lattice is a set of points in a plane that are arranged in a regular pattern. In abstract algebra, it is often represented as a group of translations in two dimensions, with a basis of two vectors that generate all the other points in the lattice.

2. What are some common problems involving plane lattices in abstract algebra?

Some common problems involving plane lattices in abstract algebra include finding the basis vectors, determining the order of the lattice, and calculating the number of points in a given area of the lattice.

3. How can I solve plane lattice problems in abstract algebra?

To solve plane lattice problems in abstract algebra, start by identifying the basis vectors of the lattice. Then, use these vectors to generate all other points in the lattice. From there, you can use algebraic operations to solve specific problems, such as finding the order or number of points in a given area.

4. What are some real-world applications of plane lattices in abstract algebra?

Plane lattices in abstract algebra have various real-world applications, such as in crystallography, where they are used to describe the arrangement of atoms in a crystal. They are also used in coding theory, where they can be used to generate error-correcting codes.

5. Are there any helpful tools or resources for solving plane lattice problems in abstract algebra?

Yes, there are various tools and resources that can aid in solving plane lattice problems in abstract algebra. These include textbooks, online tutorials, and software programs specifically designed for abstract algebra. Additionally, seeking help from a math tutor or joining a study group can also be beneficial.

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