
MAT 534 Problem Sets
Algebra I
Fall 2009

There will be weekly problem sets collected in the Thursday lecture.
Problems due one week will often practice material covered the
previous week. It is important to do the problems in a timely manner;
do not postpone problems until the last minute.
UNDER NO CIRCUMSTANCES WILL LATE HOMEWORK BE ACCEPTED.
In exceptional circumstances such as a documented medical absence,
etc., the instructor will drop the missing problem set and compute the
problem set total using the remaining problem sets.
You are encouraged to work with other students in the class, but your
final writeup must be your own and must be based on your own
understanding. Moreover, writeups must be legible, etc. Writeups
which prove too difficult for the grader to read may be marked
incorrect or may be returned to the student to be rewritten.
All questions regarding grading of a problem set should be addressed
to the instructor in a timely fashion.
In computing final grades, only the 10 best problem
sets (out of 11 total graded problem sets) will be used.
In the assignments below, please work through all problems. Only the
RED problems will be graded.
Please direct any
other questions regarding problem sets to your instructor.

Problem Set 1 is
due in the Thursday lecture the week of Sept. 7th — 11th.
Section 1.1: 9, 22.
Section 1.2: 4, 10.
Section 1.3: 2, 13.
Section 1.4: 2, 11.
Section 1.6: 9 (Hint: Use Exercise 1.2.4 and
1.3.13 assigned above.), 23.
Section 1.7: 8, 10.

Problem Set 2 is
due in the Thursday lecture the week of Sept. 14th — 18th.
Section 2.1: 6, 10.
Section 2.2: 4, 14.
Section 2.3: 21, 26.
Section 2.4: 7, 12.
Section 2.5: 11, 16.

Problem Set 3 is
due in the Thursday lecture the week of Sept. 21st — 25th.
Section 3.1: 17, 24.
Section 3.2: 4, 12.
Section 3.3: 3, 8.
Section 3.4: 7, 8.
Section 3.5: 3, 16.

Problem Set 4 is
due in the Thursday lecture the week of Sept. 28th — Oct. 2nd.
Section 3.4: 9, 10.
Section 3.5: 6, 8.
Section 4.1: 3, 7(a) and (d).
Section 4.2: 11, 12, 13.

Problem Set 5 is
due in the Thursday lecture the week of Oct. 12th — 16th.
Section 4.3: 19, 21, 24.
Section 4.4: 2, 10.
Section 4.5: 24, 38.
Section 4.6: 3, 4.

Problem Set 6 is
due in the Thursday lecture the week of Oct. 19th — 23rd.
Section 4.6: 6, 7.
Section 5.1: 7, 8*, 9*.
Section 5.2: 8, 14.
Section 5.4: 8, 9.
* There is a small mistake in Problems 8,9 of Section 5.1. Please
prove injectivity of the homomorphism from Problem 8 only for the
special case occurring in Problem 9.

Problem Set 7 is
due in the Thursday lecture the week of Oct. 26th — 30th.
Section 5.5: 7, 13, 23.
Section 11.1: 6, 8, 9.
Section 11.2: 19, 26, 33.

Problem Set 8 is
due in the Thursday lecture the week of Nov. 2nd — 6th.
Section 11.3: 2, 3, 4.
Section 11.4: 4, 5.
Section 11.5: 5, 11, 13.

Problem Set 9 is
due in the Thursday lecture the week of Nov. 9th — 13th.
Problem Set on the spectral theorem (
pdf,
dvi,
ps).

Problem Set 10 is
due in the Thursday lecture the week of Nov. 30th — Dec. 4th.
Main Problem. Let G and G'
be groups with the same finite order n.
Let m be some integer divisor of n, and assume that G and G' each
contain a unique normal subgroup P, resp. P', of order m. For each
group, consider the associated group which is the centralizer of P,
resp. P', modulo the center of P, resp. the center of P'. If G and G'
are isomorphic, prove that these associated groups are also
isomorphic.
Next, assume that there exists a subgroup Q of G which intersects P in
only the identity element and such that P and Q generate G, i.e., G is
a semidirect product of P and Q.
Let f denote the induced homomorphism from Q to the outer
automorphism group of P (not the usual homomorphism to the
automorphism group of P). Show that the kernel of f is canonically
isomorphic to the quotient of the centralizer of P by the center of
P. Thus the isomorphism class of the kernel of f is an isomorphism
invariant of G. In particular, observe that the hypotheses hold if we
restrict to groups G and G' which are each a
semidirect product of a fixed finite pgroup P and a fixed group Q
whose order is less than p+1. Moreover, if P is abelian then the
outer automorphism group of P equals the usual automorphism group of P
so that f is the usual homomorphism.
Second Problem. For every problem on Midterm 2
where you lost points, please write up a complete, correct solution to
that problem (if you only lost points on a part of the problem, you
can write up just the solution for that part). If you got full credit
on the exam, you do not need to write up anything (you will
automatically get credit for this part of the homework assignment).

Problem Set 11 is
due in the Thursday lecture the week of Dec. 7th — Dec. 11th.
This problem set is to be assigned.
Section 12.1: 16, 17, 18, 19 (Please read through this sequence of
exercises.)
Section 12.3: 2, 9, 13, 14, 17.
Back to my home page.
Jason Starr
4108 Math Tower
Department of Mathematics
Stony Brook University
Stony Brook, NY 117943651
Phone: 6316328270
Fax: 6316327631
Jason Starr