# Plane Lattices

1. Mar 7, 2010

### 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.

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