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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 Winter '08
 Rogalski,Daniel
 Math, Algebra, Complex Numbers, Subgroup, blue book

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