THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS
JUNE 2003
MATH3710
HIGHER ALGEBRA I
(1) TIME ALLOWED 3 HOURS
(2) TOTAL NUMBER OF QUESTIONS 6
(3) ATTEMPT ALL QUESTIONS
(4) THE QUESTIONS ARE NOT OF EQUAL VALUE
(5) THIS PAPER MAY BE RETAINED BY THE C
Daniel Chan
MATH3710:Assignments and Mathematical Writing
For many of you, Higher Algebra 1 will be the rst serious pure mathematics course you will
take. Unlike second year courses, the majority of questions you will be asked to do will involve
proofs (g
MATH3710: Higher Algebra 1: Group Theory (2005, S1)
Lecturer: Daniel Chan
E-Mail: danielc@unsw.edu.au
Webpage: www.maths.unsw.edu.au/danielch
Oce: Red Centre (East Wing) Room 4104
Oce Phone No.: 9385 7084
Consultation Hours: TBA (see webpage)
Most of the
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS
JULY 2004
MATH3710
HIGHER ALGEBRA I
(1) TIME ALLOWED 3 HOURS
(2) TOTAL NUMBER OF QUESTIONS 7
(3) ATTEMPT ALL QUESTIONS
(4) THE QUESTIONS ARE NOT OF EQUAL VALUE
(5) THIS PAPER MAY BE RETAINED BY THE C
Daniel Chan
MATH3710: Higher Algebra I,
Problem Sheet 3
1. Consider the subgroup R of C (you need not show it is a subgroup).
Describe geometrically, all the cosets of R in C. Identify the group
C/R i.e. show it is isomorphic to a well-known group we have
Daniel Chan
MATH3710: Higher Algebra I,
Problem Sheet 2
1. Find the subgroup of Z generated by 4 and 6.
2. Let G be the symmetric group on 4 symbols S4 and H be the subset
cfw_|(4) = 4. Show that H is a subgroup. Compute all the left
and right cosets of H
ALGEBRA 1, D. CHAN
1. Introduction
1
Introduction to groups via symmetry. A symmetry of F is a surjective isometry which preserves F .
Denition 1.1. A set G is a group when given an operation G G G, satisfying
1. associativity, (ab)c = a(bc),
2. identity,
Daniel Chan
MATH3710: Higher Algebra I,
Problem Sheet 6
1. Describe all abelian groups of order 24 up to isomorphism.
2. Let G = Z3 and H be the subgroup generated by
4
3
2 , 5 .
0
3
Write G/H as a product of cyclic groups as in lecture 23.
3. This ques
Daniel Chan
MATH3710: Higher Algebra I,
Problem Sheet 4
1. Let G = AGLn , the group of isometries on Rn and T be the subgroup of
translations. Show that T G and using the third isomorphism theorem
show that G/T is isomorphic to a well known group.
2. Let
Daniel Chan
MATH3710: Higher Algebra I,
Problem Sheet 1
1. Given the following equation in a group x1 yxz 2 = 1, solve for y.
2. Let GLn (Z) be the set of n n matrices M with integer entries such that
M 1 exists and also has integer entries. Show that GLn
Daniel Chan
MATH3710: Higher Algebra I,
Problem Sheet 5
1. Let G, H be groups. Describe the centre of GH in terms of Z(G) and
Z(H). Note that this in particular, shows that the product of abelian
groups is abelian, a fact which is easily proved directly t