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103a_05f_fs

# 103a_05f_fs - Math 103A 1(a False(f False(b True(g True...

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Math 103A Final Exam Solutions 12/5/05 1. (a) False (b) True (c) False (d) False (e) True (f) False (g) True (h) True (i) False. 2. (a) If a, b Z ( G ) and x G , then xab = axb = xab . Also ax = xa implies a - 1 xa - 1 = a - 1 xa - 1 and so xa - 1 = a - 1 x . (b) Suppose x G . Then xZ ( G ) = { xg | g Z ( G ) } = { gx | g Z ( G ) } = Z ( G ) x . 3. When multiplying permutations, odd times odd is even, odd times even is odd and even times even is even. Hence the product of the permutations α 1 , . . ., α k will be even if and only if the number of α 1 , . . ., α k which are odd is even. Since a cycle is odd if and only it has even length, this tells us that a product of cycles is even if and only if the number of even length cycles in the product is even. 4. The simplest group is D 3 . Let a and b be two different flips. All flips have order 2. Then ab negationslash = e and ab is a rotation. Hence it has order 3. Another example is S n with n 3. Then a = (12), b = (13) and ab = (132). In a sense this is the same example since we can view S 3 as a subgroup of S n and S 3 D 3 .

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