MAT 534 Problem Sets
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 write-up must be your own and must be based on your own
understanding. Moreover, write-ups must be legible, etc. Write-ups
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 (
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
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 p-group 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
Section 12.3: 2, 9, 13, 14, 17.
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