Welcome to Math 1100! (additions to this web site no longer count towards good deed points)
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Week of...
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Notes and Links
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1
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Sep 8
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About This Class; Monday - Non Commutative Gaussian Elimination; Thursday - the category of groups, automorphisms and conjugations, images and kernels.
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2
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Sep 15
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Monday - coset spaces, isomorphism theorems; Thursday - simple groups, Jordan-Holder decomposition series.
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3
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Sep 22
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Monday - alternating groups, group actions, The Simplicity of the Alternating Groups, HW1, HW 1 Solutions, Class Photo; Thursday - group actions, Orbit-Stabilizer Thm, Class Equation.
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4
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Sep 29
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Monday - Cauchy's Thm, Sylow 1; Thursday - Sylow 2.
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5
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Oct 6
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Monday - Sylow 3, semi-direct products, braids; HW2; HW 2 Solutions; Thursday - braids, groups of order 12, Braids
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6
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Oct 13
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No class Monday (Thanksgiving); Thursday - groups of order 12 cont'd.
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7
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Oct 20
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Term Test; Term Test Solutions on Monday, HW3; HW 3 Solutions; Thursday - solvable groups, rings: defn's & examples.
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8
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Oct 27
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Monday - functors, Cayley-Hamilton Thm, ideals, iso thm 1; Thursday - iso thms 2-4, integral domains, maximal ideals, One Theorem, Three Corollaries, Five Weeks
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9
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Nov 3
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Monday - prime ideals, primes & irreducibles, UFD's, Euc.Domain PID, Thursday - Noetherian rings, PID UFD, Euclidean Algorithm, modules: defn & examples, HW4, HW 4 Solutions
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10
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Nov 10
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Monday - R is a PID iff R has a D-H norm, R-modules, direct sums, every f.g. module is given by a presentation matrix, Thursday - row & column reductions plus, existence part of Thm 1 in 1t3c5w handout.
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11
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Nov 17
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Monday-Tuesday is UofT's Fall Break, HW5, Thursday - 1t3c5w handout cont'd, JCF Tricks & Programs handout
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12
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Nov 24
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Monday - JCF Tricks & Programs cont'd, tensor products, Thursday - tensor products cont'd
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13
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Dec 1
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End-of-Course Schedule; Monday - tensor products finale, extension/reduction of scalars, uniqueness part of Thm 1 in 1t3c5w, localization & fields of fractions; Wednesday is a "makeup Monday"!; Notes for Studying for the Final Exam Glossary of terms
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F
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Dec 15
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The Final Exam
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Register of Good Deeds
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![Class Photo](/images/thumb/0/03/14-1100-ClassPhoto.jpg/310px-14-1100-ClassPhoto.jpg) Add your name / see who's in!
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![10-1100-Splash.png](/images/thumb/d/de/10-1100-Splash.png/310px-10-1100-Splash.png) See Non Commutative Gaussian Elimination
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In Preparation
The information below is preliminary and cannot be trusted! (v)
This assignment is due at class time on Thursday, October 20, 2010.
Solve the following questions
- (Selick)
- What it the least integer
for which the symmetric group
contains an element of order 18?
- What is the maximal order of an element in
? (That is, of a shuffling of the red cards within a deck of cards?)
- (Selick) Let
be a subgroup of index 2 in a group
. Show that
is normal in
.
- Let
be a permutation whose cycle decomposition consists of one 5-cycle, two 3-cycles, and one 2-cycle. What is the order of the centralizer
of
?
- (Selick) Let
be a group of odd order. Show that
is not conjugate to
unless
.
- (Dummit and Foote) Show that if
is cyclic then
is Abelian.
- (Lang) Prove that if the group of automorphisms of a group
is cyclic, then
is Abelian.
- (Lang)
- Let
be a group and let
be a subgroup of finite index. Prove that there is a normal subgroup
of
, contained in
, so that
is also finite. (Hint: Let
and find a morphism
whose kernel is contained in
.)
- Let
be a group and
and
be subgroups of
. Suppose
and
. Show that ![{\displaystyle (G:H_{1}\cap H_{2})<\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/608f94e27f55cc2e4681a6c9cef542bdc35a9ad1)
Problem 1. (Selick) Show that any group of order 56 has a normal Sylow-
subgroup, for some prime
dividing 56.
Problem 2. (Qualifying exam, May 1997) Let
act on
by permuting the factors, and let
be the semi-direct product of
and
.
- What is the order of
?
- How many Sylow-5 subgroups does
have? Write down one of them.
Problem 3. (Selick) Show that the group
of unit quaternions (
, subject to
and
) is not a semi-direct product of two of its proper subgroups.
Problem 4. (Qualifying exam, September 2008) Let
be a finite group and
be a prime. Show that if
is a
-subgroup of
, then
is congruent to
mod
. You may wish to study the action of
on
by multiplication on the left.