Abstract Algebra (Math 327)
Assignments
Spring Semester, 2003
Assignments are listed by the date due.
PDF and DVI (requires
TeX software) versions of this
page are available for printing.
Most of these assignments are simply exercises designed to prepare you
for the quizzes and the written assignments. Those which are to be
submitted as written assignments are so labeled. While you may find it
helpful to discuss the exercises with others, no collaboration is permitted
on the written assignments.
 Fri., May. 9:

Final Examination, 10:30  12:30
 Question. Will the final be based on the whole course or
just on the material from after the midterm?
 Answer. The whole course with emphasis on material covered since
the midterm.
 Thu., May. 8:

Office Hours: 3:00  4:00
 Wed., May. 7:

Office Hours: 4:00  5:00
 Mon., May. 5:

Last Regular Meeting of the Course: Bring review questions.
Written Assignment No. 5 is due:
List 5 groups that represent different isomorphism classes among
the groups of order 8.
Determine the number of different ring isomorphism classes that arise
among the four rings
given as
for each of the four polynomials f(t) of degree 2 over the
field Z/2Z
Decompose the polynomial x^{12}  1 into irreducible factors
over the field Z/5Z.
Explain why every group of order 175 must contain a nontrivial
proper normal subgroup. Can a group of order 175 contain a
nonnormal subgroup?
Write a proof of the following proposition:

Proposition.
A finite abelian group of order n is cyclic if and only if for
each integer k > 0 dividing n there is one and only one
order k subgroup of G.
In your argument you may feel free to cite the theorem stating
that every finite abelian group is isomorphic to the direct
product of cyclic groups whose orders successively divide
each other.
 Fri., May. 2:

(This is an exercise, not an assignment to be submitted.)
The group G = O_{3}(Z) of 3 \times 3 orthogonal integer
matrices, i.e., integer matrices inverted by their transposes,
is a group of order 48.
Verify that G has order 48.
What isomorphism class is shared by the Sylow 3subgroups of G ?
Identify a normal subgroup of index 2 in G.
Find a commutative normal subgroup of index 6 in G.
What is the largest order of any element of G ?
Determine the number of Sylow 3subgroups of G.
Determine the number of Sylow 2subgroups of G.
Determine the isomorphism class that is shared by the
Sylow 2subgroups of G.
 Wed., Apr. 30:

Read: § 4.4
Exercises:
 392: 33, 34
 222: 11, 17
 229: 4, 6
 Mon., Apr. 28:

Read: § 4.3
Exercises:
 381: 18
 392: 30, 31
 216: 12
 222: 6, 10, 12, 13
 Fri., Apr. 25:

Read: § 4.2
Exercises:
 334: 40
 373: 20
 381: 14, 15
 391: 17, 25
 216: 7
 222: 1  3, 5
 Wed., Apr. 23:

Written Assignment No. 4 is due:
Find all units in the ring (Z/7Z)
.
Decompose the polynomial x^{8}  1 into irreducible factors
in the ring (Z/2Z)
.
Let A denote the ring Z + Z SQRT{1}
of Gaussian integers. Recall that A is a Euclidean domain,
and, therefore, a principal ideal domain.
Find an element alpha \in A for which
alpha A = Z 2 + Z (5  SQRT{1}) .

Let m >= 0 be an integer, and let R denote the
ring Z + Z SQRT{5}. Let T_{m} denote the additive
subgroup of R given by
T_{m} = Z 7 + Z (m  SQRT{5}) .

Find the smallest value of m >= 0 for which T_{m}
is an ideal in R.
Find the isomorphism class of the quotient ring R / T_{m}
for the value of m obtained in the previous part.
 Wed., Apr. 16:

This reading assignment returns to the theory of groups:
Read: § 4.1
Exercises:
 334: 37, 38
 373: 13, 15, 16
 380: 8  10
 390: 4, 6, 13
 215: 1, 3
 Mon., Apr. 14:

Read: § 8.1
Exercises:
 333: 31, 32, 34
 342: 24, 26  28
 366: 28, 30, 31
 373: 9, 12
 380: 5, 7
 390: 1  3
 Fri., Apr. 11:

Read: § 7.3
Exercises:
 307: 30  32
 333: 27  30
 342: 18, 19, 21  23
 366: 18  21, 25, 26
 373: 7
 380: 1  3
 Wed., Apr. 9:

Read: § 7.2
Exercises:
 296: 28, 29
 307: 23, 24, 27  29
 333: 20, 21, 23, 26
 342: 15  17
 366: 12, 14
 373: 1  3
 Mon., Apr. 7:

Read: § 7.1
Exercises:
 296: 24, 25, 27
 307: 16, 25, 26
 332: 15, 17, 18
 341: 7, 8, 14
 366: 1  6, 9, 10
Note: Due to icy conditions on April 4, the date for resubmission
of problem 2 from Written Assignment No. 3 is postponed until Monday,
April 7.
Also: A unannounced quiz that had been scheduled for April 4 has
been postponed until Monday, April 7.
 Fri., Apr. 4:

Read: § 6.2
Exercises:
 295: 12  14, 16, 20, 22, 23
 306: 2, 7, 9, 14
 331: 5  9, 12  14
 341: 2, 4, 5
 Wed., Apr. 2:

Read: § 5.6
Written Assignment No. 3 is due:
Find a familiar group that is isomorphic to the quotient group
of the group D_{4} (the group of symmetries of the square) by
its center.
Find a simple realvalued function f(x, y) of two variables
for which the map phi : \cal{H}_{3}(R) > \cal{H}_{3}(R) defined by
phi(x, y, t) = (u y, x, t + f(x,y))

is an automorphism of \cal{H}_{3}(R) when u = 1,
but show that for every function f(x, y) this formula does
not yield an automorphism of \cal{H}_{3}(R) when u = 1.
 Mon., Mar. 31:

Read: §§ 5.5, 6.1
Exercises:
 284: 12, 14, 15
 295: 1  11
 331: 1  3
 Fri., Mar. 28:

Read: § 5.4
Exercises:
 269: 24
 276: 19, 20, 24, 27
 283: 1, 2, 4
 Wed., Mar. 26:

Read: § 5.3
Exercises:
 262: 32, 40, 42, 46  48, 50, 52
 269: 18, 21, 23, 28
 276: 2, 3, 5, 6  9, 12, 14, 15
 and the following:
Without consulting the text write a proof of:

Proposition.
If G is a group, X a Gset, g an element of G, and
x an element of X, then there is a relation of conjugacy
between the isotropy group of the action of G on X at x and
the isotropy group of the action at g . x:
 Mon., Mar. 24:

Read: § 5.2
Exercises:
 261: 21  28, 33, 35, 37, 38
 268: 1  8, 11, 15  17
 and the following:
Let G = SL_{2}(R) be the group of all 2 \times 2 real matrices
for which det(M) = 1, i.e., a d  b c = 1.
Let H be the upperhalf plane, i.e., the set of all complex numbers
z = x + i y with y > 0 where i = SQRT{1}.
H is a Gset under the action
M . z = {a z + b}/{c z + d} .

Find the subgroup of G that acts trivially on H, i.e.,
the set of all g \in G such that g . z = z for all
z \in H.
Find the isotropy group G_{i} at i, i.e., the subgroup
of all g \in G such that g . i = i.
Find the orbit in H of i, i.e., the set of all points in
H of the form g . i for at least one
g \in G. Hint: Consider the two special cases
(1) g = 
 (when r > 0 ) and g = 
 .

 Fri., Mar. 21:

Read: § 5.1
Exercises:
 260: 1  20
 and the following concerning the 3dimensional Heisenberg group
\cal{H}_{3}(R), which is given by a “twisted” group law on
R^{3}:
(x_{1}, y_{1}, t_{1}) * (x_{2}, y_{2}, t_{2}) = (x_{1} + x_{2}, y_{1} + y_{2}, t_{1} + t_{2} + x_{1} y_{2}) .

Find the center of \cal{H}_{3}(R).
Find the commutator xi eta xi^{1} eta^{1} when
xi = (x_{1}, y_{1}, t_{1}) and eta = (x_{2}, y_{2}, t_{2}).
 Wed., Mar. 19:

Exercises:
 138: 39, 40, 42, 46, 49
 189: 39, 40
 204: 18  20
 208: 1
 Mon., Mar. 17:

Midterm Test
 Fri., Mar. 14:

Review Session: Bring questions.
 Wed., Mar. 12:

Read: §§ 3.5, 3.6
Exercises:
 135: 26  28, 30, 32, 33, 35, 44
 188: 5  9, 17  21, 24, 25, 27, 32, 34, 35
 202: 1  3, 8, 11  13, 15
 Mon., Mar. 10:

Written Assignment No. 2
For each of the following propositions either provide a proof that it
is true or provide evidence demonstrating that it is false.
Let phi : G > G' be a group homomorphism. Show that if
G is finite, then phi(G) is finite and divides
G .
Show that any group homomorphism phi : G > G' where
G is prime must either be the trivial homomorphism or else
be a onetoone map.
 Sat., Mar. 1  Sun., Mar. 9

Recess
 Fri., Feb. 28:

Finish reading § 2.4
Exercises:
 177: 36  40
 135: 9  11, 13, 14, 21, 22, 24
 188: 1  4, 14
 Wed., Feb. 26:

Read: § 3.3
Exercises:
 169: 47, 49, 50, 52
 177: 31  35
 188: 13
 Mon., Feb. 24:

Exercises:
 127: 44, 45
 169: 25  29, 44  46, 48, 51
 177: 6  8, 13  15, 17  20, 26, 27, 29, 30
 Fri., Feb. 21:

Read: § 3.2
Exercises:
 127: 37  39, 43
 169: 12  14, 21  24, 39  43
 177: 1  5, 9  12, 16, 21  25
 Wed., Feb. 19:

Written Assignment No. 1
For each of the following propositions either provide a proof that
the proposition is true or provide a counterexample demonstrating
that the proposition is false.
If the number of elements in a finite group G with identity e
is even, show that there is at least one element g in G such
that g <> e but g * g = e.
If G is a finite group and there is some nontrivial proper
subgroup H of G such that for every element g in G
one has the relation gH = Hg, then for every subgroup
H of the given group G one has gH = Hg for every g
in G.
 Mon., Feb. 17:

President's Day Recess: no class
 Fri., Feb. 14:

Read: § 3.1
Exercises:
 127: 32  34, 36
 136: 15  18
 169: 1  8, 16  20, 32, 33  38
 Wed., Feb. 12:

Read: § 2.4 through p. 132
Exercises:
 115: 11, 14, 15, 22, 23, 26, 31
 126: 16, 19  24, 26, 27, 29, 30
 135: 1  4, 7, 8
 Mon., Feb. 10:

Read: § 2.3
Exercises:
 102: 22, 28  32, 36, 45
 114: 7, 9, 10, 13
 125: 1  6, 12, 13, 15
 Fri., Feb. 7:

Read: § 2.2
Exercises:
 74: 47, 49  51
 86: 35, 39, 40  47, 52, 53, 59
 102: 11, 12, 16, 18, 20, 21
 114: 1, 2, 5
 Wed., Feb. 5:

Read: § 2.1
Exercises:
 73: 34, 36, 40, 46
 85: 18, 19, 21, 24, 25, 34, 38
 101: 1, 3, 5, 6, 7
 Mon., Feb. 3:

Read: § 1.5
Exercises:
 61: 29, 30, 33
 71: 8  13, 22, 23, 32
 85: 8  15, 17
 Fri., Jan. 31:

Read: § 1.4
Exercises:
 61: 6  9, 11  18, 23, 26, 27
 71: 1  7
 Wed., Jan. 29:

Read: Read § 1.3
Exercises:
 39: 14  22
 40: 28, 36
 48: 8  12, 18
 61: 1  5
 Mon., Jan. 27:

Reading: Scan Ch. 0; Read Ch. 1, §§ 1.1  1.2
Each of the following exercises may be done quickly if you have
paid close attention to the definitions in the reading.
Exercises:
 38: 1  13
 40: 29  32
 48: 1  7
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