Class notes
Linear algebra part 2
- Course
- Math 125
- Institution
- University Of Alberta (UofA)
Linear algebra part 2
[Show more]
Seller
Follow
Content preview
18.701 October 2003Plane crystallographic groups with point groups C4 or D4
This describes the discrete subgroups G of M whose translation group L is a lattice and whose point group
is either the cyclic group C4 or the dihedral group D4 .
The cyclic group C4 is generated by the rotation through the angle π/2 about the origin, which we denote by
ρ. In its standard representation, the dihedral group D4 contains ρ and the reflection r about the horizontal
axis. The elements of D4 are the four rotations ρi and the four reflections ρi r, where i = 0, 1, 2, 3.
Every element of the point group, in particular ρ, carries the lattice L to itself, so L is a square lattice. We
may choose coordinates in V ≈ R2 so that L becomes the lattice of vectors with integer coordinates. We
recall that
L = {a ∈ V |ta ∈ G}.
We first consider the case of a discrete subgroup H whose point group is C4 . The next proposition shows
that there is just one type of group in this case. Since ρ is in the point group H, H contains an element
of the form m = tv ρ. This element is a rotation through angle π/2 about some point of the plane P . So
H contains a rotation through angle π/2. We translate the coordinates in P so that the origin becomes the
center of the rotation m. In this new coordinate system, m = ρ, so when we write the isometries using these
coordinates, ρ ∈ H.
Proposition. Let H be a discrete subgroup of M whose translation group L is a lattice and whose point
group H is the cyclic group C4 . Then with coordinates chosen as above, H consists of the elements ta ρi ,
where a ∈ L (i.e., a is an integer vector) and i = 0, 1, 2, 3.
Proof. Let S denote the set of vectors ta ρi with a ∈ L. By definition, ta ∈ H when a ∈ L, and by our choice
of coordinates, ρ ∈ H. Therefore the products ta ρi are in H, which shows that S ⊂ H. Conversely, let h be
an element of H. Since the point group of H is cyclic, every element of H preserves orientation. So m has
the form tv ρθ for some v ∈ V and some θ. The image of m in the point group is ρθ , so θ is a multiple of
π/2, and ρθ = ρi for some i. Since H is a group, the product mρ−i = tv is in H, so v ∈ L. Therefore m ∈ S.
This shows that S ⊃ H, hence that S = H. �
We now consider the case of a discrete group G whose point group is D4 . To begin with, we note that the
orientation-preserving elements of G form a subgroup H of G of index 2. Moreover H is a discrete subgroup
of M whose translation group is L, and whose point group is C4 . With suitable choices of coordinates, H is
the group described by the previous proposition. In particular, ρ ∈ G.
Proposition. Let G be a discrete subgroup of M with point group D4 . We choose coordinates so that the
origin is a point of rotation of order four, and so that the lattice L is the integer lattice. Let U denote the
translated lattice c + L in V , where c = ( 12 , 12 )t . There are two possibilities G1 and G2 :
(a) The elements of G1 are the products ta p where a ∈ L and p ∈ D4 , or
(b) The elements of G2 are the products tv p, such that p ∈ D4 ,
v ∈ L if p is a rotation, and v ∈ U if p is a reflection.
Proof. Since the point group G contains the reflection r, G contains an element of the form m = tu r. If u
is in L, then the product t−u m = r is also in G. Because ρ ∈ G and G is a group, G contains the elements
of G1 , and an argument similar to that used in the proof of the previous proposition shows that G = G1 .
Suppose that u ∈ L. Let u = ρu. Since ρ ∈ G,
m = ρm = ρtu r = tu� r
1
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller sivachowdeswarnandipati. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for CA$11.51. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
67474 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now