NAME:
Student Number:
Oct.4, 2011
MATHEMATICS 323 - Assignment 1
Due time for this assignment: 5:00pm, Oct.14, 2011.
You may choose 5 questions. (Total: 10 marks)
1. Let G be a group and a, b G. Suppose |a| = |b| = |ab| = 2. Then show that ab = ba.
2. Let
MA323 Intro to Groups and Rings
Assignment 3
Due: Tuesday, Nov. 3rd, 2009, beginning of class
1. 8 #44.
2. What is the maximum possible order for an element of S15 ?
3. (a) Give the elements of A4 (Hint: Try using transposition notation).
(b) Find a two e
MA323 Intro to Groups and Rings
Assignment 1
Due: Thursday, Oct. 1st, 2009, beginning of class
1. 0 #18.
2. Show that a partition of a set S yields an equivalence relation on S.
3. 0 #35.
4. Is the set H = cfw_3x + 2 | x Z closed under
(a) the usual addit
MA323 Intro to Groups and Rings
Assignment 2 Answer Key
1. Determine, with proof, whether the following are groups:
(a) 4 #4.
This is not a group as the element 0 Q screws everything up. It cannot be the
identity e as 0x = x0 = 0 for all x Q but there doe
MA323 Intro to Groups and Rings
Assignment 1 Answer Key
1. 0 #18.
Let : cfw_0, 1A P(A) be dened such that, for any fB cfw_0, 1A , we have : fB B
where B = cfw_x A | fB (x) = 1. Since fB is well-dened, the set B is well-dened as
well. Thus is a well-dened
MA323 Intro to Groups and Rings
Assignment 3 Answer Key
1. 8 #44.
An n-gon has n dierent rotations, each a multiple of 360 degrees. Call these rotations
n
0 , . . . , n1 . As well, if n is even, there are n ips with axes from side to side and
2
n
2 ips wi
MA323 Intro to Groups and Rings
Assignment 2
Due: Thursday, Oct. 15th, 2009, beginning of class
1. Determine, with proof, whether the following are groups:
(a) 4 #4.
(b) 4 #18.
(c) The set of all 3 3 matrices of the form
1 a
0 1
0 0
b
c
1
with matrix mu
Fall Term, 2013
MA323 - Introduction to Groups and Rings
Instructor: Dr. Lawrence Howe
E-mail: [email protected] (Do not email me within MLS.)
Oce: BA 303D
Oce Hours: Mondays & Tuesdays, 11:30 am - 12:30 pm, or by appointment.
Lectures: TR, 10:00 - 11:20 pm, B
Fall Term, 2014
MATHEMATICS 323 Introduction to Groups and Rings
Instructor:
Office
Hours:
Dr. Kaiming Zhao (BA542)
Tuesdays and Thursdays: 11:30pm-12:30 p.m.
or by appointment
Classes: Tuesdays and Thursdays 13:00 -14:20 a.m.
Room: BA209
Text: John B. Fr
Fall Term, 2012
MATHEMATICS 323 Introduction to Groups and Rings
Instructor: Dr. Kaiming Zhao
Office: BA 542 (Bricker Academic Building)
Telephone: ext. 2444
Email: [email protected]
Office Hours: Tuesday, Thursday, 1:00 pm 2:20 pm, or by appointment
Lectures:
MA323 Midterm Summary
- Time: 1:00pm-2:20pm, Oct.23, 2014
- Room: BA209
- Total marks: 50
- 5 questions
- The test covers all material in taught in class before Oct.10
- Best way to study: study lecture notes you took in class, work out problems unfinishe
Fall Term, 2009
MATHEMATICS 323 Introduction to Groups and Rings
Instructor:
Eric Martin
Office: BA436 (Bricker Academic Building)
Telephone:
ext. 2972
Email:
[email protected]
Office Hours: Wednesday 12-2 pm, Thursday 2-4 pm, or by
appointment
Web site: htt
Fall Term, 2010
MATHEMATICS 323 Introduction to Groups and Rings
Instructor:
Dr. Kaiming Zhao
Office: BA 542 (Bricker Academic Building)
Telephone: ext. 2444
Email: [email protected]
Office Hours: Tuesday, Thursday, 1:00 pm 2:30 pm, or by appointment
Lectures:
MA323 Intro to Groups and Rings
Assignment 5
Due: Thursday, Dec. 3rd, 2009, beginning of class
1. An element a of a ring R is idempotent if a2 = a.
(a) 18 #44(a).
(b) 18 #44(b).
(c) Determine all ring homomorphisms from Z to Z.
2. (a) 18 #41.
(b) 18 #46.
MA323 Intro to Groups and Rings
Assignment 5 Answer Key
1. An element a of a ring R is idempotent if a2 = a.
(a) 18 #44(a).
Let a and b be idempotent elements of a commutative ring R. Then a2 = a and
b2 = b. Hence, since R is commutative, (ab)2 = abab = a
MA323 Intro to Groups and Rings
Assignment 4b Answer Key
1. Show that the following are rings, whether they have unity, and whether
they are elds
(a) 18 #10.
2Z is denitely an abelian subgroup of the group Z. If we notice that, for all
2a, 2b 2Z,
2a 2b =
MA323 Intro to Groups and Rings
Assignment 4a Answer Key
1. (a) Let G be a cyclic group generated by an element a and G a group.
Let : G G and : G G be group homomorphisms such that
(a) = (a). Prove that = .
Pick some x G. Since G is generated by a, we ha
MA323 Intro to Groups and Rings
Assignment 4a
Due: Tuesday, Nov. 17th, 2009, beginning of class
1. (a) Let G be a cyclic group generated by an element a and G a group. Let : G
G and : G G be group homomorphisms such that (a) = (a). Prove
that = . (This i
MA323 Intro to Groups and Rings
Assignment 4b
Due: Tuesday, Nov. 24th, 2009, beginning of class
1. Show that the following are rings, whether they have unity, and whether they are
elds:
(a) 18 #10.
(b) 18 #12.
2. 18 #20.
3. Suppose that R is a ring and th