PMath 336: Introduction to Group Theory
Exercise set 1
May 13, 2008
Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works. 1. (6 points) L

PMATH 336
Assignment 1 Solutions
Winter 2013
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1. Consider the pair (Z, ) where a b = a + b 3.
(a) Carefully prove that (Z, ) is a group. What is the identity element of the group?
Solution. Obviously Z is a nonempty set and the operation is dened for a

PMATH 336
Assignment 3 Solutions
Winter 2013
1. Let G be a nite group and p a prime number dividing |G|.
(a) Suppose H G with |H | = p. Prove that every element of H except e has order p.
Solution. Let a H . Then |a| must be a divisor of |H | = p, so |a|

PMATH 336
Assignment 4
Winter 2013
Due Friday, February 15, 2013 by 3:20 p.m.
1. (a) Show that the only subgroup of S3 containing (1 2 3) and (1 2) is S3 itself.
Solution. There are many ways to do this. For example, one can express each element of S3 as

PMath 336: Introduction to Group Theory
Midterm 3
July 14, 2008
Instructions
Solve all ve questions. Write your answers in the notebook provided. Include full arguments in the answer. Dont forget to write your name and id number on the notebook. Simple ca

PMath 336: Introduction to Group Theory
Midterm 2
June 24, 2008
Instructions
Solve all ve questions. Write your answers in the notebook provided. Include full arguments in the answer. Dont forget to write your name and id number on the notebook. Simple ca

PMath 336: Introduction to Group Theory
Midterm 1
May 30, 2008
Instructions
Write your answers in the notebook provided. Include full arguments in the answer. Dont forget to write your name and id number on the notebook. No additional material is allowed.

PMath 336: Introduction to Group Theory
Exercise set 11
July 25, 2008
Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works.
Notation
GLn

PMath 336: Introduction to Group Theory
Exercise set 10
July 25, 2008
Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works.
Notation
GLn

PMath 336: Introduction to Group Theory
Exercise set 8
July 11, 2008
Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works.
Notation
GLn (

PMath 336: Introduction to Group Theory
Exercise set 7
July 11, 2008
Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works.
Notation
GLn (

PMath 336: Introduction to Group Theory
Exercise set 5
June 20, 2008
Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works.
Notation
GLn (

PMath 336: Introduction to Group Theory
Exercise set 4
June 11, 2008
Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works.
Notation
GLn (

PMath 336: Introduction to Group Theory
Exercise set 2
May 27, 2008
Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works.
Notation
GLn (S

PMATH 336
Assignment 2 Solutions
Winter 2013
1. Suppose G is a group and a, b G.
(a) If a2 = b2 and a3 = b3 , prove that a = b.
Solution. Assume a2 = b2 and a3 = b3 . Then a = a3 (a2 )1 = b3 (b2 )1 = b.
(b) Can you think of a way to generalize the claim i