{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Week4HomeworkSupplement

# Week4HomeworkSupplement - k = ij This group has a unique...

This preview shows page 1. Sign up to view the full content.

Supplementary problem – Week 4 1. Let p be a prime number, and consider the set Z × p = { 1 , 2 , . . . , p - 1 } of nonzero residue classes modulo p . Since p is prime, these are the residue classes of integers relatively prime to p . Recall that these then form a group under multiplication. In lecture, I stated the following theorem (take Math 180 or Math 120B to see a proof): Theorem. Z × p is a cyclic group. Verify this theorem for the prime p = 11 by finding all generators for the group Z × 11 . (Hint: Find one generator by trial-and-error, and then apply Corollary 6.16.) 2. Find the subgroup of M 2 × 2 ( C ) generated by the matrices: i = 0 - 1 1 0 and j = i 0 0 - i Start by computing
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: k = ij . This group has a unique element of order 2 – ﬁnd it. What does Section 6 Exercise 50 say about this element? Remarks The group you are computing is called the group of quaternion units or sometimes just the quaternion group, and it makes a notable appearance in Math 120B. It is the smallest non-abelian group that satisﬁes the hypothesis of Section 6, Exercise 50 3. Using equation (1) on p.62 and the fact that there are inﬁnitely many prime numbers, show that the group Q of rational numbers under addition is not ﬁnitely generated....
View Full 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