Mathematics 3020A Assignment 2, due at classtime, Thursday, September 26
1. (a) Let r, s be positive integers such that ( r, s ) = 1. Prove that for every integer n 1,
( r n , sn ) = 1.
Solution: Let r, s be positive integers such that ( r, s ) = 1, and s
Assignment 1, due at classtime, Thursday, September 19
1. Let A, B be sets, and f : A B.
(a) Prove that f is injective if and only if for all S A, f 1 (f (S) = S.
Solution: Suppose that f : A B is injective, and let S A. We rst prove that S f 1 (f (S)
(an
Mathematics 3020A Assignment 3, due at classtime, Thursday, October 3
1. (a) Use the Well Ordering Principle to prove that for any positive integer a, and all nonnegative integers r, s, if ( r + 1, s + 1 ) = 1, then ( r ai , s ai ) = 1.
i=0
i=0
Solution:
Mathematics 3020A Assignment 4, due at classtime, Thursday, October 10
1. (a) Let n be a positive integer, and let Sn . For each i Jn = cfw_ 1, 2, 3 . . . , n , let
Ti = cfw_ k (i) | k Z+ . Prove that cfw_ Ti | i Jn is a partition of Jn .
Solution: For e
Mathematics 3020A Assignment 5, due at classtime, Thursday, October 17
1. The group of units U17 is cyclic. Find all generators of U17 .
Solution: We compute 3 = cfw_ 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6, 1 = Z17 cfw_ 0 = U17 . Thus 3
i
is
Mathematics 3020A Assignment 6, due at classtime, Thursday, October 24
1. Let n be a positive integer, and let k be an integer with 1 k n. Let be a k-cycle in
Sn , so there exist i0 , i1 , . . . , ik1 Jn such that = (i0 i1 . . . , ik1 ). Prove that for ev
THE UNIVERSITY OF WESTERN ONTARIO
LONDON
CANADA
DEPARTMENT OF MATHEMATICS
Math 3020A Final Exam
December 19, 2013
3 hours
1. Let G and H be groups, and let Aut(H) denote the automorphism group of H (so the
elements of Aut(H) are isomorphisms from H to H,