{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw1sol - Homework 1 Solutions 1 There are six symmetries of...

This preview shows pages 1–2. Sign up to view the full content.

Homework 1 Solutions 1. There are six symmetries of an equilateral triangle: the identity, rotate clockwise by 120 rotate counterclockwise by 120 , and three mirror symmetries about lines which pass through one vertex and the opposite edge. In class we created a partial multiplication table for these symmetries. Complete this table. Solution: Define 1 to be the identity transform, ρ to be rotation by 120 (so ρ 2 is rotation by 240 . Let σ 1 be the reflection through the vertical line (through the center), σ 2 to be the reflection through the line at 30 , and σ 3 to be reflection through the line at - 30 . Then depending on the interpretation of order, our Cayley table is one of the following: 1 ρ ρ 2 σ 1 σ 2 σ 3 1 1 ρ ρ 2 σ 1 σ 2 σ 3 ρ ρ ρ 2 1 σ 3 σ 1 σ 2 ρ 2 ρ 2 1 ρ σ 2 σ 3 σ 1 σ 1 σ 1 σ 2 σ 3 1 ρ ρ 2 σ 2 σ 2 σ 3 σ 1 ρ 2 1 ρ σ 3 σ 3 σ 1 σ 2 ρ ρ 2 1 1 ρ ρ 2 σ 1 σ 2 σ 3 1 1 ρ ρ 2 σ 1 σ 2 σ 3 ρ ρ ρ 2 1 σ 2 σ 3 σ 1 ρ 2 ρ 2 1 ρ σ 3 σ 1 σ 2 σ 1 σ 1 σ 3 σ 2 1 ρ 2 ρ σ 2 σ 2 σ 1 σ 3 ρ 1 ρ 2 σ 3 σ 3 σ 2 σ 1 ρ 2 ρ 1 2. Construct a group G with | G | = 3 (so G has exactly three elements).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern