{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}