MA5Q6 Graduate Algebra
Assignment 1
Deadline: Tuesday 16 October 2013, 12pm
Write your name and student number on your solution sheet. Mention
your department if it is not mathematics. Hand in all questions to Carole
Fishers ofce.
You are encouraged to wo
MA5Q6 Graduate Algebra
Assignment 7
Deadline: Friday 6 December 2013, 12pm
Write your name and student number on your solution sheet. Mention your department
if it is not mathematics. Hand in all questions to Carole Fishers ofce. You are encouraged
to wor
MA5Q6 Graduate Algebra
Assignment 3
Deadline: Wednesday 30 October 2013, 12pm
Write your name and student number on your solution sheet. Mention your department
if it is not mathematics. Hand in all questions to Carole Fishers ofce. You are encouraged
to
MA5Q6 Graduate Algebra
Assignment 5
Deadline: Friday 22 November 2013, 12pm
Write your name and student number on your solution sheet. Mention your department
if it is not mathematics. Hand in all questions to Carole Fishers ofce. You are encouraged
to wo
MA 5Q60
THE UNIVERSITY OF WARWICK
MSc EXAMINATION: April 2013
Graduate Algebra
Time Allowed: 3 hours
Candidates should attempt all 4 questions.
Read carefully the instructions on the answer book and make sure that the numbers
required are entered on each
MA5Q6 Graduate Algebra
Assignment 4
Deadline: Wednesday 13 November 2013, 12pm
Write your name and student number on your solution sheet. Mention your department
if it is not mathematics. Hand in all questions to Carole Fishers ofce. You are encouraged
to
/A
0
/B
/C
/0
:B!A
= idC
R
D
:B!C
c2C
b2B
(b) = (d + (a) = (d) + ( (a) = (d)
(A)
|D
D B/ (A) C
=
=
B
D \ (A) = cfw_0)
b = d + (a)
(b) = c
R
a
(b d) = (b
b = d + (a)
A
Z
C =Z
:CZ!B
=
(n) = (nb) = n (b) = n 1 = n
/A
/B
/C
0
/0
B
f 2 homR (D, A)
f 6= 0
(f )
MA5Q6 Graduate Algebra
Assignment 2
Deadline: Tuesday 22 October 2013, 12pm
Write your name and student number on your solution sheet. Mention your department
if it is not mathematics. Hand in all questions to Carole Fishers ofce. You are encouraged
to wo
G
(ab)c = a(bc) 8a, b, c 2 G
G
8a 2 G 9a
e2G
aa 1 = a
1
1
a=e
ae = ea = a 8a 2 G
N = cfw_0, 1, . . .
(N \ cfw_0, +)
(2Z, )
(Mn (2Z), )
(R, min)
(N, +)
(R [ 1, min)
G
f (ab) = f (a)f (b) 8a, b 2 G
G
f : G ! H
H
H
f (eG ) = eH
G, H
f (eG )
1
eH = f (
MA5Q6L Graduate Algebra
3
1 Introduction
1.1
Groups
Denition 1.
(a) A semigroup is a set G along with a binary operation G G G: ( a, b) ab
satisfying
associativity: a(bc) = ( ab)c for all a, b, c G.
(b) An identity element, neutral element, unit, one in a
MA5Q6L Graduate Algebra
27
Proof. Let F be a free Z-module with basis A B. Let K F be the Z-submodule
generated by X where
X = ( a + a , b) ( a, b) ( a , b) a, a A, b B
( a, b + b ) ( a, b) ( a, b ) a A, b, b B
( ar, b) ( a, rb) a A, r R, b B .
We put A
MA5Q6 Graduate Algebra
Assignment 6
Deadline: Friday 29 November 2013, 12pm
Write your name and student number on your solution sheet. Mention your department
if it is not mathematics. Hand in all questions to Carole Fishers ofce. You are encouraged
to wo