OFFICE HOURS: ZEMAN: M 12:00 -1:30, W: 12:00
- 1:30
LEWIS: Tu 1:00 - 2:00, Th 1:00 - 2:00
FINAL: Theorems for the final, Sample final , Sample solutions
REVIEW SESSION:Friday, December 5, 2:00 - 4:00pm, SSL 145
WEEK 10
M: Theorem: Every cycle from S_n can be expressed as a product
of transpositions.
Theorem: Every permutation
from S_n can be expressed as a product of transpositions.
Lemma: (a) If \sigma is
in S_n, then \sigma is its own inverse.
(b) If x,y,u,v are mutually distinct, then (x,y)(u,v)=(u,v)(x,y)
(c) If x,y,z are mutually distinct, then (y,z)(x,y)=(x,y)(x,z)
(d) If x,y,z are mutually distinct, then (x,z)(y,z)=(x,y)(x,z)
HW:
Coursebook, p.94 #7,8,9,10,11,12. Also: Check that (a) -- (d) of
the above Lemma hold.
W: Lemma: Let \sigma be a permutation from S_n. Assume that
\sigma=\tau_1\tau_2.....\tau_k
where \tau_i are transpositions. Let a be a number from the set {1,2,...,n}.
Then we
can find transpositions \pi_1,...,\pi_m such that
(a) \sigma=\pi_1\pi_2....\pi_m;
(b) \pi_1,...,\pi_{m-1} do not move a (so the only transposition in this
product that
might possibly move a is \pi_{m-1});
(c) k-m is even
Theorem: If \sigma is a premutation
from S_n and \sigma can be expressed as a product of
even number of transpositions, then it is not possible to express \sigma
as a product
of odd number of transpositions.
Signum of a permutation, sgn(\sigma).
HW:
Go over Theorems for the final, Sample
final and Sample solutions.
F: Proof of the Theorem from W using the Lemma from W.
WEEK 9
M: Equivalence relations and partitions. Orbits and cycles.
HW:
p.94 #1,3,5 and p.96 #30,30,32
W: Disjoint cycles. Theorem: Any permutation from S_n can be
expressed as a product
of disjoint cycles.
HW:
No new homework. Enjoy the holidays.
F: Holiday
WEEK 8
M: Cartesian (direct) products of proups. Projections.
The permutation group S_A.
HW:
p. 110 #1,2,3 and 4, p.113 #50 and p.134 #22
Hint for #50:
When the author says H appears as a subgroup in H x K he does NOT
literally mean H, but H x {e_K}={(a,e_K)| a is an element of H}.
Notice that this is a subgroup of H x K and is isomorphic to H. Similarly,
in the case of K he means {e_H} x K = {(e_H,b)| b is an element of K}.
Similarly, this is a subgroup of H x K and is isomorphic to K.
W: Cayley's Theorem and its proof: Every group G is isomorphic
to a subgroup of S_G.
HW:
Coursebook, p.83 #1,2,3,4,6,8 and p.86 #40,41
Tomorrow's quiz will
cover Week 7 and Monday of Week 8.
F: Completion of the proof of Cayley's Theorem.
Equivalence relations.
HW:
p.84, #18 and #20.
Also: Read the section on Equivalence relations in the Coursebook,
p.6.
WEEK 7
M: Proof of (b) from the last theorem from Friday: If
G is a cyclic group of finite order s, then G
is isomorphic to (Z_s,+_s).
Theorem: If G is a group then the intersection of any system of
subgroups of G is again
a subgroup of G. The subgroup <X> of G generated by some
subset
X of G. Theorem: The elements
of <X> are precisely all finite products of powers of
elements of X (these powers
may be negative, of course).
HW:
p68 #55 and p.72, # 1 - 6
W: Completion of the proof of the last theorem fom
Monday: if b is a finite product of integer
powers of elements of X,
then so is the inverse of b. Definition of a homomorphism.
Theorem: if f:(G_1,*_1)
--> (G_2,*_2) is a homomorphism then (a) f[H] is a subgroup
of G_2 whenever H is a subgroup
of G_1 and (b) f^{-1}[K] is a subgroub of G_1
whenever K is a subgroup
of G_2. We proved (a).
HW:
Coursebook, p.133 # 4,5, 13,14,15 and p.135 # 46 DUE:
Friday,
November 14
Regarding the Hint to #46: The theorem the book refers to is the theorem
which
I numbered by 7.2 in the lecture (and whose proof we completed today) It
says
that the elements of <X> are precisely all finite products of
powers of
elements of X .
F: Corrolary of (a) of the theorem from Wendesday: if f is a
homomorphism, then f(e) is the
identity and f(a^{-1}) is the inverse
to f(a). Proof of (b) of that theorem. The kernel Ker(f)
of a homomorphism f. Theorem: A homomorphism
f is injective if and only if Ker(f) ={e}.
HW:
Coursebook, p. 133 #17,18 , p.134 #20,25,26,27 and p135, #50
Hint: Regarding #17 - 27: Apply #46 from the last assignment
that tells us that
homomorphisms are determined by their values on the generators and
make use of the fact that all groups from the above problems are cyclic.
WEEK 6
M: Midterm
HW:
Coursebood, p.66 #12-16, p.67 #45 and #47
Hint: #47: consider the intersection of <r> and <s>.
W: Order of an element in a group. Theorem: if G=<a> then
the order of G is equal to the order
of a in G and all elements e,a,a^2,...,a^(s-1)
are all mutually distinct where s is the order of G.
Theorem: If G=<a> of order
s and d=the greatest common divisor of r,s then (a) <a^r>=<a^d>
and (b) <a^r> has s/d
many elements.
HW:
Coursebook, p67 #51 and p68, #52 and #53
F: Completion of the proof of the last theorem from Monday.
Some examples of groups and
structure of their subgroups.
Theorem: (a) If G is an infinite cyclic group, then G is isomorphic
to (Z,+). (b) If G is a
cyclic group of finite order s, then G is isomorphic to (Z_s,+_s).
HW:
No new homework. Focus on homework from Wendesday.
WEEK 5
M: A cyclic subgroup generated by a. Definition of a cyclic
group.
HW:
Coursebook, p58 #45, 52 , p. 66 #22,23 and p.67 #44
Hint: #45: To see that the condition in the exercise implies that H is
a subgroup,
use the criterion on being a subgroup. Prove that H contains the identity
and inverse elements first. Then prove that H is closed under the operation.
#52 This is similar to #51, see friday's homework.
#44 Compute \phi(a^n) and \psi(a^n) and compare them.
W: Degree of a group. Formulae for a^{m+n} and a^{m.n}. Basic
properties of cyclic groups:
any subgroup of a cyclic group
is cyclic, every cyclic group is Abelian. If G is cyclic with
a generator a and a^s=e for some
integer s, then G has at most s elements.
HW: See
the Sample Midterm (Focus on the solutions to
2,3 and 4)
F: Review for the Midterm
WEEK 4
M: A proof that (U,.) and (R,+) are non-isomorphic. A three
element group is unique up to an
isomorphism, so is isomorphic
to (Z_3,+_3). Subgroups of (Z_3,+_3). Four - element groups:
(Z_4,+_4) and its subgroups. Klein's
fourgroup V.
HW:
No new assignment today. Try to work on the problems from previous assignments
that you did not get.
W: Klein's fourgroup and its subgroups. A criterion on being
a subgroup.
HW:
p.55 #5,6, p56#15,16 and #20 DUE:Friday,
October 24
F: A criterion on being a subgroup. Cyclic subgroups.
HW: Coursebook,
p.55 #11,12, p.56 #22,23 p.58 #51 and p.59 #54
(In #11,12: det(AB)=det(A).det(B). In #51: The expression "xa" means
"x*a")
WEEK 3
M: Uniqueness of the identity element. Preservation of the identity
element under isomorphism.
The example of non-isomorphic
structures from Friday, Week 2 (Completion).
HW:
Coursebook, p.34 Exercises 16 and 17, p.35 Exercises 24 and 25, and p.36
Exercise 32
(For the last exersise: A structural property is a property which is preserved
under
isomorphisms. That is: if (S,*) and (T,.) are isomorphic, then one of them
has the
property if and only if the other does.)
W: Definition of a group. Examples of groups. Cancellation
laws.
HW:
p.45 Exercises 9,10 and p.49, exercise 41 DUE:
Friday, October 17
F: Linear equations in groups: existence and uniqueness of solutions.
Uniqueness of
inverse elements. Abelian groups. Definition
of a subgroup. Trivial subgroups.
Two element group and its subgroups.
HW: Coursebook,
p.46 Exercises 12 - 15, p.48 Exercise 30 (How many left identities
and how many right identities has this structure?) and p.49, Exercise 39
WEEK 2
M: Associativity, commutativity of binary structures.
Subsets of structures closed under operations.
Substructures. Tables
describing binary operations.
HW:
Coursebook, p. 26 Exercises 4,15,16 and p. 27 Exercises 23,26. Matrix
Multiplication
W: Isomorphism of binary structures. Isomorphism between (Z,+)
and (2Z,+). Isomorphism between
(R,+) and (R^+,.).
HW:
Coursebook, p. 28 Exercise 36 and p.36, Exercise 27 DUE:
Friday,
October 10
F: Proving that structures are non-isomorphis. Examples of non-isomorphic
pairs of
structures: (Z,+) and (Q,+), (R,.) and (M_2(R),.),
(Z,+) and (Q^+,.).
HW: Coursebook,
p34 Exercises 6,7,11,13 and p36, Exercise 26,33. Matrix
Multiplication
WEEK 1
M: The unit circle (U,.) with multiplication. The half-open
interval [0,2\pi) with addition modulo 2\pi. The
isomorphism between
the two structures.
HW:
Coursebook, p19 Exercises 16 - 28.
W: The isomorphism between (U,.) and [0,2\pi) with addition
modulo 2\pi. Roots of unity, i.e. the roots of
the equation
x^m=1.
HW:
Generalize the structure "[0,2\pi) with addition modulo 2\pi" as follows:
Given is a positive real
number c. Define
the operation "addition modulo c" on the half-open interval [0,c). Find
the neutral
element for the structure
"[0,c) with the addition modulo c". Also, for each x in [0,c) find the
inverse
element in this structure.
F: Roots of unity, the isomophism between the structure
(U_m, .) and Z_m with addition modulo m.
HW:
Coursebook, p26 Exercise 12,17 - 22 and 24a,g,h,i,j.
WEEK 0
F: Examples of algebraic structures (Z,+), (Q,.), (C,.). Multiplication
of complex numbers and polar coordinates.
HW: Coursebook,
p19 Exercises 6,7,8,9,10,11,12,13.