MAT-4720 Midterm: MATH 4720 App State Fall2007 Exam1
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Answer key: (20 points): if the statement is always true, circle true and prove it. If the statement is never true, circle false and prove that it can never be true. If the statement is true in some cases and false in others, circle possible and give an example and a counter-example. (a) let g g such that g12 = 1. If |g| = 5, then g5 = 1 so that g12 = g5 g5 g2 = 1 1 g2 = g2 = 1. But this means that |g| 2 < 5 = |g| contradiction! (b) de ne : g g by (x) = x 1. True / possible / false: is an automorphism of g. Each element has a unique inverse so that is a bijection. Multiply this equation by ab on the left and ba on the right and get: ba = ab. Therefore, is an automorphism if and only if g is abelian.
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