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Week4HomeworkSupplement

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

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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
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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....
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