UNIVERSITY OF TORONTO
The Faculty of Arts and Sciences
DECEMBER 2014 EXAMINATION PERIOD
MAT301H1F
Instructor: Ali Mousavidehshikh
NAME:
STUDENT NUMBER:
Instructions:
This exam consists of 8 questions for a total of 100 marks, and 10 pages including the c

MAT301 ASSIGNMENT 4 SOLUTIONS
DUE DATE: FEBRUARY 8, 2017, AT THE BEGINNING OF YOUR TUTORIAL
Question 1. Suppose G is a group with the following three properties:
G is Abelian,
|G| = 35,
x35 = e for all x G.
Prove that G is cyclic.
Proof. Given a G, a35

Lecture 2 Mao Dun and
Shen Congwen
C. T. Hsia: Obsession with
China
Modern Chinese literatures [1912-1949]
obsessive concern with China as a nation
afflicted with a spiritual disease and therefore
unable to strengthen itself or change its set
ways of inh

Lecture 1 Advance to the
modern, and Lu Xun
Li Ruzhen (1763-1830)
Jinghuayuan Romance of the Flowers
in the Mirror (1810-1820); 1st ed. 1828
Some buds of modernity in Flowers in
the Mirror
Sea of Regret
Arthur H. Smith (1945-1932)
Chinese Characterist

MAT301 ASSIGNMENT 2
DUE DATE: WEDNESDAY JUNE 1, 2016 AT THE BEGINNING OF YOUR TUTORIAL
Question 1. Let
a b
GL(2, R) =
: a, b, c, d R, ad bc 6= 0
c d
As mentioned in class, GL(2, R) is a group under matrix multiplication (you do not have to prove this).
De

MAT301 ASSIGNMENT 4 SOLUTIONS
DUE DATE: FRIDAY FEBRUARY 26TH, AT THE BEGINNING OF YOUR TUTORIAL
For all questions show your work. Unless otherwise stated G is always a group.
Question 1. (a) Suppose a, b G with |a| = |b| = p, where p is a prime number. Sh

UNIVERSITY OF TORONTO
The Faculty of Arts and Sciences
SUMMER 2015 EXAMINATION PERIOD
MAT301H1Y
Instructor: Ali Mousavidehshikh
NAME:
STUDENT NUMBER:
Instructions:
This exam consists of 9 questions for a total of 100 marks, and 10 pages including the cov

MAT301 ASSIGNMENT 6 SOLUTIONS
DUE DATE: FRIDAY APRIL 1, 2016, AT THE BEGINNING OF YOUR TUTORIAL
For all the questions show your work. There are 7 questions on this assignment, you should
do them all. However, only three of them will be marked.
Question 1.

MAT301 ASSIGNMENT 2 SOLUTIONS
DUE DATE: FRIDAY JANUARY 22ND, AT THE BEGINNING OF YOUR TUTORIAL
For all questions show your work.
Question 1. (a) Given a natural number n 2, define
U (n) = cfw_[x] : gcd(x, n) = 1, 1 x n
where [x] is the equivalence class o

MAT301 ASSIGNMENT 1
DUE DATE: FRIDAY JANUARY 15TH, AT THE BEGINNING OF YOUR TUTORIAL
Question 1. Let H be a non-empty set and define G = cfw_f : H H : f bijection (that is, G is the set of
all bijections from H to itself). Prove that (G, ) is a group (whe

MAT301 ASSIGNMENT 5 SOLUTIONS
DUE DATE: FRIDAY MARCH 4, 2016, AT THE BEGINNING OF YOUR TUTORIAL
For all questions show your work.
Question 1. Show that : Z12 Z10 (both groups are under addition) defined via (x) = 3x is not a
group homomorphism.
Solution.

MAT301 ASSIGNMENT 5
DUE DATE: FRIDAY MARCH 4, 2016, AT THE BEGINNING OF YOUR TUTORIAL
For all questions show your work.
Question 1. Show that : Z12 Z10 (both groups are under addition) defined via (x) = 3x is not a
group homomorphism.
Question 2. (a) If G

MAT301 ASSIGNMENT 6
DUE DATE: FRIDAY APRIL 1, 2016, AT THE BEGINNING OF YOUR TUTORIAL
For all the questions show your work. There are 7 questions on this assignment, you should
do them all. However, only three of them will be marked.
Question 1. Prove tha

MAT301 ASSIGNMENT 1 SOLUTIONS
DUE DATE: FRIDAY JANUARY 15TH, AT THE BEGINNING OF YOUR TUTORIAL
Question 1. Let H be a non-empty set and define G = cfw_f : H H : f bijection (that is, G is the set of
all bijections from H to itself). Prove that (G, ) is a

MAT301 ASSIGNMENT 4
DUE DATE: FRIDAY FEBRUARY 26TH, AT THE BEGINNING OF YOUR TUTORIAL
For all questions show your work. Unless otherwise stated G is always a group.
Question 1. (a) Suppose a, b G with |a| = |b| = p, where p is a prime number. Show that if

MAT301 ASSIGNMENT 2
DUE DATE: FRIDAY JANUARY 22ND, AT THE BEGINNING OF YOUR TUTORIAL
For all questions show your work.
Question 1. (a) Given a natural number n 2, define
U (n) = cfw_[x] : gcd(x, n) = 1, 1 x n
where [x] is the equivalence class of x mod n.

MAT301 ASSIGNMENT 3
DUE DATE: FRIDAY JANUARY 29TH, AT THE BEGINNING OF YOUR TUTORIAL
For all questions show your work.
Question 1. Suppose |G| = 3. Prove that G is cyclic.
Question 2. Let G be a group. For any a G and H G, define
aH = cfw_ah : h H.
Suppos