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103a_05f_1e - |(This can be done without doing(a but it is...

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Math 103A Midterm Exam (50 points) 26 October 2005 Please put your name and ID number on your blue book. The exam is CLOSED BOOK except for one page of notes. Calculators are NOT allowed. In a multipart problem, you can do later parts without doing earlier ones. You must show your work to receive credit. 1. (10 pts.) If S is a subset of the complex numbers, let S * be the nonzero numbers in S . Recall that Z are the integers and Q are the rationals. Answer the following TRUE or FALSE. (a) Z with addition is a subgroup of Q with addition. (b) Q * with multiplication is a subgroup of Q with addition. (c) Z * with multiplication is a subgroup of Q * with multiplication. (d) For all n > 0, the even permutations in S n form a group. (e) For all n > 0, the odd permutations in S n form a group. 2. (8 pts.) Let α = (1543)(235) be an element of S 5 . (a) Write α as a product of disjoint cycles. (b) Compute the order of α ; that is, compute | α
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Unformatted text preview: | . (This can be done without doing (a), but it is easier if you do (a).) 3. (16 pts.) Let n > 0 be an integer and let G be an abelian group. De±ne the set G n by G n = { g n | g ∈ G } . (a) Prove that G n is a subgroup of G . (b) Let D 3 be the dihedral group of rotations and re²ections of an equilateral triangle. Show that ( D 3 ) 3 is not a subgroup of D 3 . Hint : What elements of D 3 are in ( D 3 ) 3 ? 4. (16 pts.) Let G be the set of complex numbers of absolute value 1. Recall from 20B : (You may ±nd this helpful.) Complex numbers of absolute value 1 in polar coordinates have r = 1 and arbitrary angle θ . They can be written as e θi . Also e 2 πi = 1 and e ri e si = e ( r + s ) i . (a) Prove that G is a group under multiplication. (b) For all integers n > 0, ±nd a subgroup of G of order n . END OF EXAM...
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