Help with Plane Lattices Problems in Abstract Algebra

[Proggy99]

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$$^n$$ 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$$^-^1$$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.

 

[mighty2000]

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 =