Assignment 1 (due April 13th): page
22, questions 9,12 and 15; page 28, questions 6 and 15; Solutions.
Assignment 2 (due April 20th): page
33, questions 1 and 2; page 34, questions 20; page 40, questions 4,5
and 6. Solutions.
Assignment 3 (due April 27th): page
48, questions 2, 3, 4 and 10; page 60, questions 1, 2 and 3 ( in
question #3 on page 60, |x| means the
first positive power of x which realizes the
neutral element) Solutions.
Assignment 4 (due May 4th): page
44-45, questions 2,3 and 16; page 60, questions 10, 11; page 95,
questions 8. (Hint for the last question: use the fact that if
(m,n)=d, then there exists integers a, b such that d=an+bm.)
Assignment 5: No assignment for this
Assignment 6 (due May 18th): page 85,
question 3; page 88, questions 22 and 24; page 96, question 18.
Assignment 7 (due May 25th): page 101,
questions 3 and 7; page 165, question 2(a)(c), page 166 questions
4(a)(b) and 9. Solutions.
Assignment 8 (due on or before Tuesday
June 6th): Write a short expository essay (2-3 pages, typed or
handwritten) on one of the following three topics:
For the first topic, you should discuss first the
notion of action as well as the associated concepts (orbit,
stabilizer group, etc). Present
then the statement and detailed proof of
Lagrange's Theorem and discuss its importance in group theory. For
the second topic, you should
give the exact statement and proof of Burnside's
Lemma and discuss in details at least one counting application of
it. For the third topic,
you should start by introducing the notions of
p-subgroups and Sylow p-subgroups. State the three Sylow theorems
and present a
detailed proof for one of them. Give then an
example of an application of the Sylow's theorems to the
classification of finite groups.
This is a writing assignment, so the main focus
should be on clearly communicating the ideas in the proof. I
recommend looking at the
textbook (or your favorite mathematics texts) and
trying to emulate their style.