MATH 335 001 Computational Algebra Winter 2006
Assignment 3.2 GROUPS
1) Use Tietze transformations to show that the group G = x, y, z | (xy )2 xy 2 . is a free group of rank 2. 2) Show that the groups given by the following presentations x, y, z |x = yzy
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Assignment 2
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your grade. We recommend that you hand it in after you complete Unit 2. You must
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Assignment 3
This assignment, based on the content of Unit 3 of the Study Guide, is worth 3% of
your grade. We recommend that you hand it in after you complete Unit 3. You must
show all of your work in order to obtain full marks. For your convenience, eac
Assignment 4
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your grade. We recommend that you hand it in after you complete Unit 4. You must
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Assignment 5
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your grade. We recommend that you hand it in after you complete Unit 5. You must
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MATH 335 001 Computational Algebra Winter 2006
Assignment 3.1 GROUPS
1) Find a complete rewriting system for the group G = x, y, z | x3 = 1, y 2 = 1, xy = yx2 , z 2 = 1, xz = zx, zy = yz . Show that |G| = 12. 2) Show that the group H = x, y | xn = 1, y 2
MATH 335 001 Computational Algebra Winter 2006
Assignment 2.1. Monoids.
1) Let A = cfw_a1 , . . . , an be a nite alphabet and A the free monoid with basis A. For a given word u A nd all solutions in A of the equation xu = ux. 2) Let M1 = a, b | abab = ,
MATH 335 001 Computational Algebra Winter 2006
Assignment 2. Part 1.
1) Find a polynomial bijective enumeration : N N N. [Hint: Look at the Cantors enumeration of pairs] 2) a) Let N1 = N, +, 0 be the systems of natural numbers equipped with addition. Writ
MATH 335 001 Computational Algebra Winter 2006
Assignment 1. Part 1.
1) Dene by recursion the function x xm , x, m N and specify precisely the functions f and g from the corresponding recursion scheme. 2) Show that the function xy = x y, 0, if x y, otherw
MATH 335 001 Computational Algebra Winter 2006
Assignment 4.1 Grobner Basis Solve the following problems from the notes on Grobner basis: 1) 1.5.4 (page 32). 2) 1.5.5 (page 32). 3) 1.6.1 (page 36). 4) Let G and G be two Grobner bases for an ideal I k [X ]