MATH 223 Assignment 5
due March 29
45 marks + 2 bonus
1. (4 marks) Let f = (1, 6, 3)(4, 5)(2, 7, 8) and g = (3, 5, 2, 6, 7)(4, 8) nd gf g 1 and f gf 1 .
gf g 1 = (g(1), g(6), g(3)(g(4), g(5)(g(2), g(7), g(8) = (1, 7, 5)(8, 2)(6, 3, 4)
and
f gf 1 = (f (3),
MATH 223 Assignment 4
due March 1
35 marks + bonus
1. (4 marks) Let G be a group and a, b G. Prove that a and bab1 have
the same order. Give an example of a group G and elements a and b
where a and bab1 generate dierent groups.
Assume that the order of a
MATH 223 Assignment 5
due March 8
35 marks + 5 bonus
1. (12 marks) Determine if the following pairs of groups are isomorphic
by either dening an isomorphism and proving that your isomorphism
is indeed an isomorphism or showing that no isomorphism is possi
MATH 223 Assignment 2
due January 26
45 marks+ 5 bonus
1. Dene Z to be the set of all elements in Zn that have multiplicative
n
inverses.
(a) (2 marks) List the elements of Z .
8
The elements of Z are [1], [3], [5], [7].
8
(b) (3 marks) Write the multipli
MATH 223 Assignment 3 - soln
due February 2
30 marks
1. (3 marks) Let G be a group. Assume that for all x, y G that
(xy)2 = x2 y 2 . Show that G is abelian.
We need to show that for all x, y G that xy = yx. We know that
xyxy = xxyy
if we multiply on the l
MATH 223 Assignment 7
due April 10
35 marks
1. (5 marks) Let R and S be arbitrary rings. In the Cartesian product
R S dene the following operations:
(a) (r, s) = (r , s ) if and only if r = r and s = s ,
(b) (r, s) + (t, u) = (r + t, s + u),
(c) (r, s) (t
MATH 223, Introduction to Abstract Algebra, Winter 2012
Test 1, February 14, 2012,
Prof. Karen Meagher
FAMILY NAME:
FIRST NAME:
Possible Points
ID:
Actual Points:
PLEASE READ THESE INSTRUCTIONS CAREFULLY.
1. This test has 6 questions. Each question has
1. (a) (4 marks) Let A is a set with a binary operation 4:. Show that if the operation a: has an identity,
(hen tnathe identity is unique. .
f
. . 'r J
(b) (3 marks) Is thef'operation 0n ZS dened by .1: * y = :1; associative? Justify your answer!
(0) (3
MATH 223 Assignment 1
due January 19
40 marks
1. (2 marks) Dene a binary operation on A = cfw_1, 2, 3 that is commutative but not associative. (Give a specic example of elements a, b, c
with (a b) c = a (b c).)
There are lots of dierent examples but this