MAT301 ASSIGNMENT 2
DUE DATE: JUNE 4, 2014 AT THE BEGINNING OF YOUR TUTORIAL
Question 1. Give an example of a set G and an operation on G such that is not a binary operation on
G, but, the following properties hold;
(1) For all a, b, c G, we have a (b c)
SYLLABUS FOR MAT301H- GROUPS AND SYMMETRIES
SUMMER 2014
Instructor: Ali Mousavidehshikh
Lecture: Wednesday 6-9pm, MP103
Oce: BA6191 (Bahen Center)
Email: [email protected]
Webpage: Blackboard (portal)
Oce hours: Wednesdays 12-2 or by ap
MAT301 ASSIGNMENT 1 SOLUTIONS
DUE DATE: MONDAY MAY 25TH, 2015 AT THE BEGINNING OF LECTURE
Question 1. (a) Define the following relation on the integers:
x y 5|(2x + 3y)
Prove that this is an equivalence relation. Find the equivalence classes [0] and [1].
MAT301 ASSIGNMENT 2 SOLUTIONS
DUE DATE: MONDAY JUNE 8TH, 2015 AT THE BEGINNING OF LECTURE
Question 1. Let
a b
: a, b, c, d R, ad bc 6= 0
G = GL(2, R) =
c d
Note:
ad bc = det
a b
c d
(a) Prove that G is a group under matrix multiplication (G is called
MAT301 ASSIGNMENT 1
DUE DATE: SEPTEMBER 24, 2014 AT THE BEGINNING OF LECTURE (6:10PM)
Problem 1. Let G = Q cfw_0, and define a ? b =
Show that (G, ?) is a group.
ab
(where ab is the usual product of rational numbers).
2
Solution. Given a, b G, a, b are no
MAT301 ASSIGNMENT 6
DUE DATE: NOVEMBER 26, 2014 AT THE BEGINNING OF YOUR TUTORIAL
Question 1. Let G be a finite group.
(a) Let H and K be subgroups of G with H K G. Prove [G : H] = [G : K][K : H].
Solution. Since |G| < , we have |H|, |K| < . Moreover, H K
MAT301 ASSIGNMENT 3
DUE DATE: OCTOBER 8, 2014 AT THE BEGINNING OF LECTURE (6:10 PM)
Question 1. Let G be a group and H G. For any element g G, define gH = cfw_gh| h H.
(a) Suppose G is abelian and |g| = 2. Show that K G, where K = H gH.
Solution. Since e
MAT301 ASSIGNMENT 2
DUE DATE: JUNE 4, 2014 AT THE BEGINNING OF YOUR TUTORIAL
Question 1. Give an example of a set G and an operation on G such that is not a binary operation on
G, but, the following properties hold;
(1) For all a, b, c G, we have a (b c)
MAT301 ASSIGNMENT 5
DUE DATE: JULY 23, 2014 AT THE BEGINNING OF YOUR TUTORIAL
Question 1. (a) Let G be a group. Prove Inn(G) Aut(G).
Solution. Let eG be the identity element in G. Then the map e : G G given via e(g) = g for all g G is
the identity element
MAT301 ASSIGNMENT 3
DUE DATE: JUNE 11, 2014 AT THE BEGINNING OF YOUR TUTORIAL
Question 1. Let G be a group and H G. For any fixed g G, define gHg 1 = cfw_ghg 1 | h H and
NG (H) = cfw_g G| gHg 1 = H.
(a) Prove gHg 1 G for all g G.
Solution. Let g be any el
MAT301 ASSIGNMENT 4
DUE DATE: JULY 9, 2014 AT THE BEGINNING OF YOUR TUTORIAL
Question 1. (a) Let G be a group. For each a G, define clG (a) = cfw_xax1 | x G. Prove that these
subsets of G partition G (we defined partitions in the first class, its definiti
MAT301 ASSIGNMENT 6
DUE DATE: AUGUST 6, 2014 AT THE BEGINNING OF YOUR TUTORIAL
Question 1. (a) Show that G H is Abelian if and only if G and H are Abelain.
Solution. Suppose G and H are Abelian. Given x, y G H, then x = (g, h), y = (a, b) for some
g, a G,
UNIVERSITY OF TORONTO
The Faculty of Arts and Sciences
MAT301F MIDTERM
OCTOBER 15,2014
Instructor: Ali Mousavidehshikh
NAME:
STUDENT NUMBER:
Instructions:
This midterm consists of 6 questions for a total of 100 marks, and 9 pages including the cover
page
MAT301 ASSIGNMENT 3
DUE DATE: JUNE 11, 2014 AT THE BEGINNING OF YOUR TUTORIAL
Question 1. Let G be a group and H
NG (H) = cfw_g G| gHg 1 = H.
(a) Prove gHg 1
G. For any xed g G, dene gHg 1 = cfw_ghg 1 | h H and
G for all g G.
Solution. Let g be any elemen
MAT301 ASSIGNMENT 1
DUE DATE: MAY 21, 2014
Problem 1. Determine if the following sets under the given operations form a group or not. If it is a group,
prove it, if not, give an example where one of the axioms fails.
(a) The set of odd integers under addi
MAT301 Term Test
Wednesday June 18, 2014
NAME:
STUDENT NUMBER:
Question 1
/10
Question 2
/15
Question 3
/15
Question 4
/15
Question 5
/15
Question 6
/30
Total
/100
1
MAT301H1Y
Page 2 of 6
Question 1. (10 marks) Let G be a group and H a cyclic subgroup of
MAT301 Term Test
Wednesday June 18, 2014
NAME:
STUDENT NUMBER:
Question 1
/10
Question 2
/15
Question 3
/15
Question 4
/15
Question 5
/15
Question 6
/30
Total
/100
1
MAT301H1Y
Page 2 of 6
Question 1. (10 marks) Let G be a group and H a cyclic subgroup of
MAT301 ASSIGNMENT 5
DUE DATE: NOVEMBER 12, 2014 AT THE BEGINNING OF LECTURE (6:10 PM)
Question 1. (a) Is U (20) isomorphic to U (24) (explain your answer)?
Solution. Notice that
U (20) = cfw_1, 3, 7, 9, 11, 13, 17, 19, U (24) = cfw_1, 5, 7, 11, 13, 17, 19
MAT301 ASSIGNMENT 1
DUE DATE: MAY 21, 2014
Problem 1. Determine if the following sets under the given operations form a group or not. If it is a group,
prove it, if not, give an example where one of the axioms fails.
(a) The set of odd integers under addi
MAT301 ASSIGNMENT 4
DUE DATE: OCTOBER 29TH, 2014 AT THE BEGINNING OF LECTURE (6:10 PM)
Question 1. Show that any group of order 4 is abelian.
Solution. Let G = cfw_e, a, b, c where e is the identity element of G and the elements are all distinct (since
G
MAT301 ASSIGNMENT 2
DUE DATE: OCTOBER 1ST, 2014 AT THE BEGINNING OF LECTURE (6:10PM)
Question 1. Given a group (G, ), for each a G define
CG (a) = cfw_g G | g a = a g
(a) Prove that CG (a) G for all a G.
Solution. Given a G, notice that e a = a e e CG (a)
MAT301 ASSIGNMENT 5
DUE DATE: JULY 23, 2014 AT THE BEGINNING OF YOUR TUTORIAL
Question 1. (a) Let G be a group. Prove Inn(G)
Aut(G).
(b) Find Aut(Z), where Z is a group under addition (explain your answer).
(c) Show that R and Q are not isomorphic (under