I'm currently a senior in physics. I have an amateur interest in mathematics. I'm looking for an accesible introduction to the topic of Algebra
https://cameroncounts.wordpress.com/2016/11/26/a-crash-course-on-group-theory/
A crash course on group theory
2016-11-26
None
<div><div>
<div>
<p>
I have just finished a crash course on group theory at Universidade de Lisboa. The notes are <a href="https://cameroncounts.files.wordpress.com/2016/11/groups.pdf">here</a>.
</p>
<p>
From the preface:
</p>
<div>
<p>
On a visit to Universidade de Lisboa in November 2016, I was asked to give a “crash course” in group theory. The only specifications were that the course should cover both finite and infinite groups and should be accessible to students.
</p>
<p>
This is a tall order. I have tried to meet it by starting at the beginning, moving fairly fast, omitting many proofs (this means leaving many proofs to the reader). But I hope the result is still of some use, so I am making the notes of the course available.
</p>
<p>
There is a clear focus in the chapter on finite groups: we want to be able to describe them all. The <em>Jordan–Hölder theorem</em> reduces the problem to describing the finite simple groups and how general groups are built out of simple groups: the <em>Classification of Finite Simple Groups</em> solves the first part; a discussion of groups of prime power order shows that we cannot expect a nice solution to the second.
</p>
<p>
For infinite groups, such a focus is much more difficult to obtain. There is no general theory of infinite groups, and group theorists have imposed various finiteness conditions on their groups. I discuss, somewhat in the manner of a tourist guide, free groups, presentations of groups, periodic and locally finite groups, residually finite and profinite groups, and my own interest, oligomorphic permutation groups.
</p>
<p>
I thank the students for their interest and their questions.
</p>
</div>
</div>
<div>
<p> </p>
<div>
<h2>
	About Peter Cameron	</h2><p>
	I count all the things that need to be counted.	</p>
</div>
</div>
<div><p>
	This entry was posted in <a href="https://cameroncounts.wordpress.com/category/lecture-notes/">Lecture notes</a> and tagged <a href="https://cameroncounts.wordpress.com/tag/classification-of-finite-simple-groups/">Classification of Finite Simple Groups</a>, <a href="https://cameroncounts.wordpress.com/tag/finite-groups/">finite groups</a>, <a href="https://cameroncounts.wordpress.com/tag/first-order-logic/">first-order logic</a>, <a href="https://cameroncounts.wordpress.com/tag/groups-of-prime-power-order/">groups of prime power order</a>, <a href="https://cameroncounts.wordpress.com/tag/infinite-groups/">infinite groups</a>, <a href="https://cameroncounts.wordpress.com/tag/jordan-holder-theorem/">Jordan-Holder Theorem</a>, <a href="https://cameroncounts.wordpress.com/tag/locally-finite-groups/">locally finite groups</a>, <a href="https://cameroncounts.wordpress.com/tag/oligomorphic-permutation-groups/">oligomorphic permutation groups</a>, <a href="https://cameroncounts.wordpress.com/tag/periodic-groups/">periodic groups</a>, <a href="https://cameroncounts.wordpress.com/tag/profinite-groups/">profinite groups</a>, <a href="https://cameroncounts.wordpress.com/tag/residually-finite-groups/">residually finite groups</a>, <a href="https://cameroncounts.wordpress.com/tag/sylows-theorem/">Sylow's theorem</a>, <a href="https://cameroncounts.wordpress.com/tag/topology/">topology</a>. Bookmark the <a href="https://cameroncounts.wordpress.com/2016/11/26/a-crash-course-on-group-theory/">permalink</a>.	</p></div>
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 A crash course on group theory

 Posted on 26/11/2016 by Peter Cameron

 I have just finished a crash course on group theory at Universidade de Lisboa. The notes are here.

 From the preface:

 On a visit to Universidade de Lisboa in November 2016, I was asked to give a “crash course” in group theory. The only specifications were that the course should cover both finite and infinite groups and should be accessible to students.

 This is a tall order. I have tried to meet it by starting at the beginning, moving fairly fast, omitting many proofs (this means leaving many proofs to the reader). But I hope the result is still of some use, so I am making the notes of the course available.

 There is a clear focus in the chapter on finite groups: we want to be able to describe them all. The Jordan–Hölder theorem reduces the problem to describing the finite simple groups and how general groups are built out of simple groups: the Classification of Finite Simple Groups solves the first part; a discussion of groups of prime power order shows that we cannot expect a nice solution to the second.

 For infinite groups, such a focus is much more difficult to obtain. There is no general theory of infinite groups, and group theorists have imposed various finiteness conditions on their groups. I discuss, somewhat in the manner of a tourist guide, free groups, presentations of groups, periodic and locally finite groups, residually finite and profinite groups, and my own interest, oligomorphic permutation groups.

 I thank the students for their interest and their questions.

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 This entry was posted in Lecture notes and tagged Classification of Finite Simple Groups, finite groups, first-order logic, groups of prime power order, infinite groups, Jordan-Holder Theorem, locally finite groups, oligomorphic permutation groups, periodic groups, profinite groups, residually finite groups, Sylow's theorem, topology. Bookmark the permalink.
 ← Michel Deza
 Road closures and idempotent-generated semigroups →

 1 Response to A crash course on group theory

 1. Shahrooz Janbaz says:
 26/11/2016 at 10:45

 Thanks dear professor for this sharing. I will study it asap.

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I have just finished a crash course on group theory at Universidade de Lisboa. The notes are here.
From the preface:
On a visit to Universidade de Lisboa in November 2016, I was asked to give a “crash course” in group theory. The only specifications were that the course should cover both finite and infinite groups and should be accessible to students.
This is a tall order. I have tried to meet it by starting at the beginning, moving fairly fast, omitting many proofs (this means leaving many proofs to the reader). But I hope the result is still of some use, so I am making the notes of the course available.
There is a clear focus in the chapter on finite groups: we want to be able to describe them all. The Jordan–Hölder theorem reduces the problem to describing the finite simple groups and how general groups are built out of simple groups: the Classification of Finite Simple Groups solves the first part; a discussion of groups of prime power order shows that we cannot expect a nice solution to the second.
For infinite groups, such a focus is much more difficult to obtain. There is no general theory of infinite groups, and group theorists have imposed various finiteness conditions on their groups. I discuss, somewhat in the manner of a tourist guide, free groups, presentations of groups, periodic and locally finite groups, residually finite and profinite groups, and my own interest, oligomorphic permutation groups.
I thank the students for their interest and their questions.
 
	About Peter Cameron	 
	I count all the things that need to be counted.	
 
	This entry was posted in Lecture notes and tagged Classification of Finite Simple Groups, finite groups, first-order logic, groups of prime power order, infinite groups, Jordan-Holder Theorem, locally finite groups, oligomorphic permutation groups, periodic groups, profinite groups, residually finite groups, Sylow's theorem, topology. Bookmark the permalink.
A crash course on group theory
https://cameroncounts.wordpress.com/2016/11/26/a-crash-course-on-group-theory/
https://www.cantorsparadise.com/an-invitation-to-group-theory-c81e21ab739a
Introduction to Group Theory - Cantor’s Paradise
2019-03-18
Panda the Red
<div><div><div><div><div><a href="https://medium.com/@notaredpanda?source=post_page-----c81e21ab739a--------------------------------"><div><p></p></div></a><a href="https://www.cantorsparadise.com/?source=post_page-----c81e21ab739a--------------------------------"><div><p></p></div></a></div><div><div><p></p></div><div><p></p></div></div></div><div><div></div><div></div></div></div><p>Imagine a square of paper lying flat on your desk. I ask you to close your eyes. You hear the paper shift. When you open your eyes the paper doesn’t appear to have changed. What could I have done to it while you weren’t looking?</p><p>It’s obvious that I haven’t rotated the paper by 30 degrees, because then the paper would look different.</p><p>I also did not flip it over across a line connecting, say, one of the corners to the midpoint of another edge. The paper would look different if I had.</p></div></div>||||I|||| Open in app

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 Imagine a square of paper lying flat on your desk. I ask you to close your eyes. You hear the paper shift. When you open your eyes the paper doesn’t appear to have changed. What could I have done to it while you weren’t looking?

 It’s obvious that I haven’t rotated the paper by 30 degrees, because then the paper would look different.

 I also did not flip it over across a line connecting, say, one of the corners to the midpoint of another edge. The paper would look different if I had.

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Imagine a square of paper lying flat on your desk. I ask you to close your eyes. You hear the paper shift. When you open your eyes the paper doesn’t appear to have changed. What could I have done to it while you weren’t looking? It’s obvious that I haven’t rotated the paper by 30 degrees, because then the paper would look different. I also did not flip it over across a line connecting, say, one of the corners to the midpoint of another edge. The paper would look different if I had.
Introduction to Group Theory - Cantor’s Paradise
https://www.cantorsparadise.com/an-invitation-to-group-theory-c81e21ab739a
https://canvas.harvard.edu/courses/30943
MATH 122: Algebra I: Theory of Groups and Vector Spaces
2024-03-27
None
<div><div>
<p></p>
<p></p>
<p><strong>Announcements</strong></p>
<p></p>
<p></p>
<p></p>
<p></p>
<p></p>
<p></p>
<p></p>
<p></p>
<p></p>
<p><strong>Lectures, Notes, and Resources</strong></p>
<p><strong> <a href="https://docs.google.com/document/d/1LSJ1oMp4G8FfOUDutTiJoZyko4Qhm9fi0OifNphM9U0/edit?usp=sharing">You can ask in-class questions, or questions in general, here.</a></strong></p>
<p>Here is a running list of past and upcoming lecture topics (in case you want to read ahead in some of the suggested sources in the syllabus). Chances are I will not cover every topic I want to cover in the next lecture, so topics will start bleeding from one lecture to the next. (Hence only a seven-day forecast at most.) </p>
<ol>
<li>Wed, Aug 30. First Class. Groups as symmetries. Definitions groups, homomorphisms and isomorphisms. Examples: Z, R, R-0, GLn(R), Sn. <a href="http://math.harvard.edu/~hirolee/pdfs/2017-fall-122-01-groups-homorphisms-examples.pdf">Notes by Nat</a>. See Artin Chapter 2.</li>
<li>Fri, Sep 1. Basic facts about groups. Group tables. Z/nZ, Abelian-ness, mattress group. [Not covered; will cover later: Basic facts about homomorphisms, subgroups, subgroups of Z. Cyclic groups. Cyclic notation.] 
</li>
<li>Wed, Sep 6. Group actions. Orbits. More examples as needed; perhaps counting formula. (Toward Orbit-stabilizer.)
</li>
<li>Fri, Sep 8. Hiro is out of town; guest lecture. Equivalence relations, equivalence classes. Challenge problems. <a href="https://canvas.harvard.edu/courses/30943/files/4461383/download?verifier=rFzULNx7txABRn8nK9AoaUxLoDQZXojb5zAlBlht&amp;wrap=1">Here is the hand-out.</a> See Artin 2.7. 
</li>
<li>Mon, Sep 11. Lagrange's Theorem. Normal subgroups and quotient groups. Artin 2.8 and 2.12 and 6.7. <a href="https://canvas.harvard.edu/courses/30943/files/4492970/download?verifier=v2k68lmxohPozesxnks82ykBz5goOKBHJMH63VIg&amp;wrap=1">Nat's Notes.</a> [Not covered: All topics listed.]</li>
<li>Wed, Sep 13. Normal subgroups and quotient groups.  Artin 1.5. <a href="https://canvas.harvard.edu/courses/30943/files/4527914/download?verifier=3sVJNdvkkkPrPPCFNLGiAXzenPgEV3NMQtkckxsA&amp;wrap=1">Notes by Nat.</a>  [Topics not covered: Symmetric group. Cyclic notation. Cyclic groups. p-groups.]</li>
<li>Fri, Sep 15. Quotient groups. Kernels, images, first isomorphism theorem, universal property of quotient groups. [Topics not covered: Conjugation action. Class equation. Artin 7.2.] <a href="https://canvas.harvard.edu/courses/30943/files/4529538/download?verifier=m30SDMlmfKm9FoD7j1F3vsz6tIZ6DLe4f9TkW94Y&amp;wrap=1">Notes by Nat.</a>
</li>
<li>Mon, Sep 18. First isomorphism theorem, universal property of quotient groups. Cyclic groups. Subgroup generated by a single element, order of elements. Classification fo cyclic groups. [Not covered: Centers. Commutator subgroup. Abelianizations.] (Artin 7.10, 2.12, 2.4) <a href="https://canvas.harvard.edu/courses/30943/files/4537483/download?verifier=9It697tr545BgrhWr1JR0lg1Qk65U6FNIjPvDsgJ&amp;wrap=1">Notes by Nat.</a>
</li>
<li>Wed, Sep 20. Dihedral group. Generators and relations. 
</li>
<li>Fri, Sep 22. S_n and A_n. Sign. Subgroups and diagrams of subgroups. Conjugation in S_n. (Artin 7.5, 1.5, 2.2)</li>
<li>Mon, Sep 25. Conjugation in S_n. Cayley's Theorem. Abelianizations, commutator subgroups. [Topics not covered: Short exact sequences. Classifying finite groups. Simple groups.] (Artin 2.4, 6.1, exercises for 6.8)</li>
<li>Wed, Sep 27. More on quotients; what it means to generate; forget cosets. [Not covered: A_n. Index. Automorphisms of groups. Conjugation action. Class equation. Orbit-stabilizer theorem, counting. Centers of p-groups.] (Special Homework Assignment: 20 Questions for Groups.)</li>
<li>Fri, Sep 29. Conjugation. Automorphisms of groups. Class equation. Orbit-stabilizer theorem, counting. Centers of p-groups. Cauchy's Theorem, index (i.e., applications of counting).[Artin 6.1]</li>
<li>Mon, Oct 2. Midterm One given out. Example Week begins: Fundamental groups.</li>
<li>Wed, Oct 4. Example week continues: Elliptic curves.</li>
<li>Fri, Oct 6. Example week continues: Subgroup of O(3), or symmetries of regular polyhedra in R^3.</li>
</ol>
<ul>
<li>Mon, Oct 9. No class!</li>
</ul>
<ol>
<li>Wed, Oct 11. Midterm due. Deadline for Classroom-to-table. Subgroups of SO(3).</li>
<li>Fri, Oct 13. Equivalence relations. Introduction to rings.</li>
<li>Mon, Oct 16. Fields, units. Z/nZ. More rings. Endomorphisms of abelian groups as rings. [not covered: Zero divisors.] (Artin Chapter 11)</li>
<li>Wed, Oct 18. Ring homomorphisms. Ideals. Quotient rings. [not covered: Zero divisors. Fields.] (Artin Chapter 11)</li>
<li>Fri, Oct 20. Z/nZ, Z/pZ, integral domains, fields, examples. Modules. Linear algebra over arbitrary fields. (Artin Chapters 11 and 14)</li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
<li></li>
</ol>
<ul>
<li></li>
</ul>
<ol>
<li>Sometime in the future: What can we do with integers? Rings, subrings of C, modules. (Artin Chapter 11)</li>
<li>Direct products and semidirect products. Short exact sequences.</li>
<li>Classification of finite groups. Simple groups. Jordan-Holder decomposition.</li>
<li>
</li>
<li>
</li>
<li>
</li>
</ol>
<p>There are also <a href="https://drive.google.com/open?id=0BzsKH9xBcZkdMlF6ckhoeC1VeFE">live-TeXed notes</a> taken by Daniel.</p>
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 MATH 122: Algebra I: Theory of Groups and Vector Spaces

 Jump to Today

 Times: MWF 2 PM
 Location: Northwest Building B101
 Instructor: Hiro Lee Tanaka

 Hiro's Office: Science Center 341 (in the back of the Birkhoff Math Library)

 Here is the syllabus: 2017-fall-122-syllabus.pdf

 Here are office hours and sections.

 Here are solutions to past homeworks (updated irregularly).

 The notes from the past rendition of my Math 122 class are here .

 This course is an introduction to groups and rings, which are foundational concepts in modern mathematics. Along the way, we will also deepen our understanding of linear algebra and the role of structures on vector spaces.

 Prerequisites: I will assume you are all familiar with real vector spaces, linear transformations, and matrices. Basic familiarity with mathematical proof is necessary; I recommend that you have taken at least one proof-based class before.

 Announcements

 Tue, Sep 12. Office hours and sections are being finalized. Follow the link for the schedule.

 Tue, Sep 12. See announcement about new room location: Northwest building Room B101. This applies for tomorrow (Wed, Sep 13).

 Mon, Sep 11. See announcement about clarifications on optional problems for HW 2.

 Sat, Sep 9. Please see announcement about how to submit homeworks.

 Friday Sept 1, 10 PM: There was ambiguity in problem 3(b); there, "id_Z" means the identity function from Z to itself. It does NOT mean "identity element of Z." The updated pdf file for the homework reflects this in blue font.

 Friday Sept 1, 10 PM: The notes from the past rendition of my Math 122 class are here .

 Friday Sept 1, 10 PM: Please be patient as your CAs and I figure out the best way to split up the grading. It seems easiest to break up the assignments problem by problem, so each CA can grade a different problem without crossing wires. It's okay if your uploaded PDFs/images contain superfluous problems (e.g., if what you upload to "Problem Two" also includes your solutions to Problems One or Three, for instance).

 Your first two homework assignments are posted: One assignment is a survey, the other is a problem set. Both are due by Wed, Sep 6.

 Finally, feel free to check out Beyond this course , where I have small links to topics/history/ideas that this course has influenced, or has been influenced by.

 Lectures, Notes, and Resources

 You can ask in-class questions, or questions in general, here.

 Here is a running list of past and upcoming lecture topics (in case you want to read ahead in some of the suggested sources in the syllabus). Chances are I will not cover every topic I want to cover in the next lecture, so topics will start bleeding from one lecture to the next. (Hence only a seven-day forecast at most.)

 1. Wed, Aug 30. First Class. Groups as symmetries. Definitions groups, homomorphisms and isomorphisms. Examples: Z, R, R-0, GLn(R), Sn. Notes by Nat. See Artin Chapter 2.
 2. Fri, Sep 1. Basic facts about groups. Group tables. Z/nZ, Abelian-ness, mattress group. [Not covered; will cover later: Basic facts about homomorphisms, subgroups, subgroups of Z. Cyclic groups. Cyclic notation.] See Artin Chapter 2. Notes by Nat. Addendum from Hiro .
 3. Wed, Sep 6. Group actions. Orbits. More examples as needed; perhaps counting formula. (Toward Orbit-stabilizer.) See Artin 6.7. Notes by Nat.
 4. Fri, Sep 8. Hiro is out of town; guest lecture. Equivalence relations, equivalence classes. Challenge problems. Here is the hand-out. See Artin 2.7. How was the exercise session? Fill out this survey!
 5. Mon, Sep 11. Lagrange's Theorem. Normal subgroups and quotient groups. Artin 2.8 and 2.12 and 6.7. Nat's Notes. [Not covered: All topics listed.]
 6. Wed, Sep 13. Normal subgroups and quotient groups. Artin 1.5. Notes by Nat. [Topics not covered: Symmetric group. Cyclic notation. Cyclic groups. p-groups.]
 7. Fri, Sep 15. Quotient groups. Kernels, images, first isomorphism theorem, universal property of quotient groups. [Topics not covered: Conjugation action. Class equation. Artin 7.2.] Notes by Nat.
 8. Mon, Sep 18. First isomorphism theorem, universal property of quotient groups. Cyclic groups. Subgroup generated by a single element, order of elements. Classification fo cyclic groups. [Not covered: Centers. Commutator subgroup. Abelianizations.] (Artin 7.10, 2.12, 2.4) Notes by Nat.
 9. Wed, Sep 20. Dihedral group. Generators and relations. (Artin 7.2, exercises for 7.10) [Topics not covered: S_n, Centers, commutator subgroups, abelianizations. Quaternions, dihedral groups. Conjugation action. Class equation.] Notes by Nat.
 10. Fri, Sep 22. S_n and A_n. Sign. Subgroups and diagrams of subgroups. Conjugation in S_n. (Artin 7.5, 1.5, 2.2)
 11. Mon, Sep 25. Conjugation in S_n. Cayley's Theorem. Abelianizations, commutator subgroups. [Topics not covered: Short exact sequences. Classifying finite groups. Simple groups.] (Artin 2.4, 6.1, exercises for 6.8)
 12. Wed, Sep 27. More on quotients; what it means to generate; forget cosets. [Not covered: A_n. Index. Automorphisms of groups. Conjugation action. Class equation. Orbit-stabilizer theorem, counting. Centers of p-groups.] (Special Homework Assignment: 20 Questions for Groups.)
 13. Fri, Sep 29. Conjugation. Automorphisms of groups. Class equation. Orbit-stabilizer theorem, counting. Centers of p-groups. Cauchy's Theorem, index (i.e., applications of counting).[Artin 6.1]
 14. Mon, Oct 2. Midterm One given out. Example Week begins: Fundamental groups.
 15. Wed, Oct 4. Example week continues: Elliptic curves.
 16. Fri, Oct 6. Example week continues: Subgroup of O(3), or symmetries of regular polyhedra in R^3.
 * Mon, Oct 9. No class!
 1. Wed, Oct 11. Midterm due. Deadline for Classroom-to-table. Subgroups of SO(3).
 2. Fri, Oct 13. Equivalence relations. Introduction to rings.
 3. Mon, Oct 16. Fields, units. Z/nZ. More rings. Endomorphisms of abelian groups as rings. [not covered: Zero divisors.] (Artin Chapter 11)
 4. Wed, Oct 18. Ring homomorphisms. Ideals. Quotient rings. [not covered: Zero divisors. Fields.] (Artin Chapter 11)
 5. Fri, Oct 20. Z/nZ, Z/pZ, integral domains, fields, examples. Modules. Linear algebra over arbitrary fields. (Artin Chapters 11 and 14)
 6. Mon, Oct 23. Free modules. Submodules. Ideals are submodules. Maps of modules. Kernel, Cokernel, Image, Matrices. Determinants. Generation of modules. Generation of ideals.
 7. Wed, Oct 25. Adjugate matrices; invertible linear transformations; Cayley-Hamilton Theorem. GL_n(F)
 8. Fri, Oct 27. Matrices are the same thing as endomorphisms of R^{\oplus n}. [Not covered: Polynomials with coefficients in fields. Integers. PIDs. Irreducibility. Euclidean algorithm for polynomials. Quotients of polynomial rings.]
 9. Mon, Oct 30. Matrices with unit determinants are invertible. [Not covered: Classification of finitely generated modules over PIDs. Classification of abelian groups.]
 10. Wed, Nov 1. Finitely generated modules over vector spaces are free. [Not covered: Back to groups. p-Sylow subgroups. Sylow theorems. Applications.]
 11. Fri, Nov 3. Cayley-Hamilton Theorem. [Not covered: Short exact sequences. Semidirect products.] Some notes about, including a proof of, the Cayley-Hamilton Theorem.
 12. Mon, Nov 6. (If we're not behind: Review session.)
 13. Wed, Nov 8. (If we're not behind: Review session.)
 14. Fri, Nov 10. In-class midterm.
 15. Mon, Nov 13. Sylow theorems.
 16. Wed, Nov 15. Applications of Sylow theorems. Cauchy's theorem.
 17. Fri, Nov 17. Semi-direct products.
 18. Mon, Nov 20. Statement of classification of finitely generated modules over PIDs.
 19. Mon, Nov 27. Application of classification of finitely generated modules over PIDs. Polynomials. Jordan Normal Form.
 20. Wed, Nov 29. Final Exam distributed. Lower central series. Why the classification of finite simple groups?
 21. Fri, Dec 1. Last Day of classes.
 * Thu, Dec 14. Final Exam due.

 1. Sometime in the future: What can we do with integers? Rings, subrings of C, modules. (Artin Chapter 11)
 2. Direct products and semidirect products. Short exact sequences.
 3. Classification of finite groups. Simple groups. Jordan-Holder decomposition.
 4. 
 5. 
 6. 

 There are also live-TeXed notes taken by Daniel.

 Course Summary:

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 Course assignments are not weighted.

Announcements 
 
 
 
 
 
 
 
 
 
 Lectures, Notes, and Resources 
  You can ask in-class questions, or questions in general, here.  
Here is a running list of past and upcoming lecture topics (in case you want to read ahead in some of the suggested sources in the syllabus). Chances are I will not cover every topic I want to cover in the next lecture, so topics will start bleeding from one lecture to the next. (Hence only a seven-day forecast at most.) 
Wed, Aug 30. First Class. Groups as symmetries. Definitions groups, homomorphisms and isomorphisms. Examples: Z, R, R-0, GLn(R), Sn. Notes by Nat. See Artin Chapter 2.
Fri, Sep 1. Basic facts about groups. Group tables. Z/nZ, Abelian-ness, mattress group. [Not covered; will cover later: Basic facts about homomorphisms, subgroups, subgroups of Z. Cyclic groups. Cyclic notation.] 
Wed, Sep 6. Group actions. Orbits. More examples as needed; perhaps counting formula. (Toward Orbit-stabilizer.)
Fri, Sep 8. Hiro is out of town; guest lecture. Equivalence relations, equivalence classes. Challenge problems. Here is the hand-out. See Artin 2.7. 
Mon, Sep 11. Lagrange's Theorem. Normal subgroups and quotient groups. Artin 2.8 and 2.12 and 6.7. Nat's Notes. [Not covered: All topics listed.]
Wed, Sep 13. Normal subgroups and quotient groups.  Artin 1.5. Notes by Nat.  [Topics not covered: Symmetric group. Cyclic notation. Cyclic groups. p-groups.]
Fri, Sep 15. Quotient groups. Kernels, images, first isomorphism theorem, universal property of quotient groups. [Topics not covered: Conjugation action. Class equation. Artin 7.2.] Notes by Nat.
Mon, Sep 18. First isomorphism theorem, universal property of quotient groups. Cyclic groups. Subgroup generated by a single element, order of elements. Classification fo cyclic groups. [Not covered: Centers. Commutator subgroup. Abelianizations.] (Artin 7.10, 2.12, 2.4) Notes by Nat.
Wed, Sep 20. Dihedral group. Generators and relations. 
Fri, Sep 22. S_n and A_n. Sign. Subgroups and diagrams of subgroups. Conjugation in S_n. (Artin 7.5, 1.5, 2.2)
Mon, Sep 25. Conjugation in S_n. Cayley's Theorem. Abelianizations, commutator subgroups. [Topics not covered: Short exact sequences. Classifying finite groups. Simple groups.] (Artin 2.4, 6.1, exercises for 6.8)
Wed, Sep 27. More on quotients; what it means to generate; forget cosets. [Not covered: A_n. Index. Automorphisms of groups. Conjugation action. Class equation. Orbit-stabilizer theorem, counting. Centers of p-groups.] (Special Homework Assignment: 20 Questions for Groups.)
Fri, Sep 29. Conjugation. Automorphisms of groups. Class equation. Orbit-stabilizer theorem, counting. Centers of p-groups. Cauchy's Theorem, index (i.e., applications of counting).[Artin 6.1]
Mon, Oct 2. Midterm One given out. Example Week begins: Fundamental groups.
Wed, Oct 4. Example week continues: Elliptic curves.
Fri, Oct 6. Example week continues: Subgroup of O(3), or symmetries of regular polyhedra in R^3.
Mon, Oct 9. No class!
Wed, Oct 11. Midterm due. Deadline for Classroom-to-table. Subgroups of SO(3).
Fri, Oct 13. Equivalence relations. Introduction to rings.
Mon, Oct 16. Fields, units. Z/nZ. More rings. Endomorphisms of abelian groups as rings. [not covered: Zero divisors.] (Artin Chapter 11)
Wed, Oct 18. Ring homomorphisms. Ideals. Quotient rings. [not covered: Zero divisors. Fields.] (Artin Chapter 11)
Fri, Oct 20. Z/nZ, Z/pZ, integral domains, fields, examples. Modules. Linear algebra over arbitrary fields. (Artin Chapters 11 and 14)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Sometime in the future: What can we do with integers? Rings, subrings of C, modules. (Artin Chapter 11)
Direct products and semidirect products. Short exact sequences.
Classification of finite groups. Simple groups. Jordan-Holder decomposition.
There are also live-TeXed notes taken by Daniel.
MATH 122: Algebra I: Theory of Groups and Vector Spaces
https://canvas.harvard.edu/courses/30943
https://www.khanacademy.org/math/algebra
Algebra 1 | Math | Khan Academy
2023-01-01
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 Combining like terms : Algebra foundationsIntroduction to equivalent expressions : Algebra foundationsDivision by zero : Algebra foundations

 Solving equations & inequalities

 Linear equations with variables on both sides : Solving equations & inequalitiesLinear equations with parentheses : Solving equations & inequalitiesAnalyzing the number of solutions to linear equations : Solving equations & inequalities
 Linear equations with unknown coefficients : Solving equations & inequalitiesMulti-step inequalities : Solving equations & inequalitiesCompound inequalities : Solving equations & inequalities

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 Rate conversion : Working with unitsAppropriate units : Working with unitsWord problems with multiple units : Working with units

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 x-intercepts and y-intercepts : Linear equations & graphsApplying intercepts and slope : Linear equations & graphsModeling with linear equations and inequalities : Linear equations & graphs

 Forms of linear equations

 Intro to slope-intercept form : Forms of linear equationsGraphing slope-intercept equations : Forms of linear equationsWriting slope-intercept equations : Forms of linear equations
 Point-slope form : Forms of linear equationsStandard form : Forms of linear equationsSummary: Forms of two-variable linear equations : Forms of linear equations

 Systems of equations

 Introduction to systems of equations : Systems of equationsSolving systems of equations with substitution : Systems of equationsSolving systems of equations with elimination : Systems of equations
 Equivalent systems of equations : Systems of equationsNumber of solutions to systems of equations : Systems of equationsSystems of equations word problems : Systems of equations

 Inequalities (systems & graphs)

 Checking solutions of two-variable inequalities : Inequalities (systems & graphs)Graphing two-variable inequalities : Inequalities (systems & graphs)Modeling with linear inequalities : Inequalities (systems & graphs)

 Functions

 Evaluating functions : FunctionsInputs and outputs of a function : FunctionsFunctions and equations : FunctionsInterpreting function notation : FunctionsIntroduction to the domain and range of a function : FunctionsDetermining the domain of a function : Functions
 Recognizing functions : FunctionsMaximum and minimum points : FunctionsIntervals where a function is positive, negative, increasing, or decreasing : FunctionsInterpreting features of graphs : FunctionsAverage rate of change : FunctionsAverage rate of change word problems : FunctionsIntro to inverse functions : Functions

 Sequences

 Introduction to arithmetic sequences : SequencesConstructing arithmetic sequences : SequencesIntroduction to geometric sequences : Sequences
 Constructing geometric sequences : SequencesModeling with sequences : SequencesGeneral sequences : Sequences

 Absolute value & piecewise functions

 Graphs of absolute value functions : Absolute value & piecewise functionsPiecewise functions : Absolute value & piecewise functions

 Exponents & radicals

 Exponent properties review : Exponents & radicalsRadicals : Exponents & radicalsSimplifying square roots : Exponents & radicals

 Exponential growth & decay

 Exponential vs. linear growth : Exponential growth & decayExponential expressions : Exponential growth & decayGraphs of exponential growth : Exponential growth & decay
 Exponential vs. linear growth over time : Exponential growth & decayExponential growth & decay : Exponential growth & decayExponential functions from tables & graphs : Exponential growth & decayExponential vs. linear models : Exponential growth & decay

 Quadratics: Multiplying & factoring

 Multiplying monomials by polynomials : Quadratics: Multiplying & factoringMultiplying binomials : Quadratics: Multiplying & factoringSpecial products of binomials : Quadratics: Multiplying & factoringIntroduction to factoring : Quadratics: Multiplying & factoring
 Factoring quadratics intro : Quadratics: Multiplying & factoringFactoring quadratics by grouping : Quadratics: Multiplying & factoringFactoring quadratics with difference of squares : Quadratics: Multiplying & factoringFactoring quadratics with perfect squares : Quadratics: Multiplying & factoringStrategy in factoring quadratics : Quadratics: Multiplying & factoring

 Quadratic functions & equations

 Intro to parabolas : Quadratic functions & equationsSolving and graphing with factored form : Quadratic functions & equationsSolving by taking the square root : Quadratic functions & equationsVertex form : Quadratic functions & equationsSolving quadratics by factoring : Quadratic functions & equationsThe quadratic formula : Quadratic functions & equations
 Completing the square intro : Quadratic functions & equationsMore on completing the square : Quadratic functions & equationsStrategizing to solve quadratic equations : Quadratic functions & equationsQuadratic standard form : Quadratic functions & equationsFeatures & forms of quadratic functions : Quadratic functions & equationsComparing quadratic functions : Quadratic functions & equationsTransforming quadratic functions : Quadratic functions & equations

 Irrational numbers

 Mastery unavailable
 Irrational numbers : Irrational numbersSums and products of rational and irrational numbers : Irrational numbersProofs concerning irrational numbers : Irrational numbers

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 Not feeling ready for this? Check out Get ready for Algebra 1 .
 * Course summary
 Algebra foundations
 Overview and history of algebra : Algebra foundationsIntroduction to variables : Algebra foundationsSubstitution and evaluating expressions : Algebra foundations
 Combining like terms : Algebra foundationsIntroduction to equivalent expressions : Algebra foundationsDivision by zero : Algebra foundations
 Solving equations & inequalities
 Linear equations with variables on both sides : Solving equations & inequalitiesLinear equations with parentheses : Solving equations & inequalitiesAnalyzing the number of solutions to linear equations : Solving equations & inequalities
 Linear equations with unknown coefficients : Solving equations & inequalitiesMulti-step inequalities : Solving equations & inequalitiesCompound inequalities : Solving equations & inequalities
 Working with units
 Rate conversion : Working with unitsAppropriate units : Working with unitsWord problems with multiple units : Working with units
 Linear equations & graphs
 Two-variable linear equations intro : Linear equations & graphsSlope : Linear equations & graphsHorizontal & vertical lines : Linear equations & graphs
 x-intercepts and y-intercepts : Linear equations & graphsApplying intercepts and slope : Linear equations & graphsModeling with linear equations and inequalities : Linear equations & graphs
 Forms of linear equations
 Intro to slope-intercept form : Forms of linear equationsGraphing slope-intercept equations : Forms of linear equationsWriting slope-intercept equations : Forms of linear equations
 Point-slope form : Forms of linear equationsStandard form : Forms of linear equationsSummary: Forms of two-variable linear equations : Forms of linear equations
 Systems of equations
 Introduction to systems of equations : Systems of equationsSolving systems of equations with substitution : Systems of equationsSolving systems of equations with elimination : Systems of equations
 Equivalent systems of equations : Systems of equationsNumber of solutions to systems of equations : Systems of equationsSystems of equations word problems : Systems of equations
 Inequalities (systems & graphs)
 Checking solutions of two-variable inequalities : Inequalities (systems & graphs)Graphing two-variable inequalities : Inequalities (systems & graphs)Modeling with linear inequalities : Inequalities (systems & graphs)
 Functions
 Evaluating functions : FunctionsInputs and outputs of a function : FunctionsFunctions and equations : FunctionsInterpreting function notation : FunctionsIntroduction to the domain and range of a function : FunctionsDetermining the domain of a function : Functions
 Recognizing functions : FunctionsMaximum and minimum points : FunctionsIntervals where a function is positive, negative, increasing, or decreasing : FunctionsInterpreting features of graphs : FunctionsAverage rate of change : FunctionsAverage rate of change word problems : FunctionsIntro to inverse functions : Functions
 Sequences
 Introduction to arithmetic sequences : SequencesConstructing arithmetic sequences : SequencesIntroduction to geometric sequences : Sequences
 Constructing geometric sequences : SequencesModeling with sequences : SequencesGeneral sequences : Sequences
 Absolute value & piecewise functions
 Graphs of absolute value functions : Absolute value & piecewise functionsPiecewise functions : Absolute value & piecewise functions
 Exponents & radicals
 Exponent properties review : Exponents & radicalsRadicals : Exponents & radicalsSimplifying square roots : Exponents & radicals
 Exponential growth & decay
 Exponential vs. linear growth : Exponential growth & decayExponential expressions : Exponential growth & decayGraphs of exponential growth : Exponential growth & decay
 Exponential vs. linear growth over time : Exponential growth & decayExponential growth & decay : Exponential growth & decayExponential functions from tables & graphs : Exponential growth & decayExponential vs. linear models : Exponential growth & decay
 Quadratics: Multiplying & factoring
 Multiplying monomials by polynomials : Quadratics: Multiplying & factoringMultiplying binomials : Quadratics: Multiplying & factoringSpecial products of binomials : Quadratics: Multiplying & factoringIntroduction to factoring : Quadratics: Multiplying & factoring
 Factoring quadratics intro : Quadratics: Multiplying & factoringFactoring quadratics by grouping : Quadratics: Multiplying & factoringFactoring quadratics with difference of squares : Quadratics: Multiplying & factoringFactoring quadratics with perfect squares : Quadratics: Multiplying & factoringStrategy in factoring quadratics : Quadratics: Multiplying & factoring
 Quadratic functions & equations
 Intro to parabolas : Quadratic functions & equationsSolving and graphing with factored form : Quadratic functions & equationsSolving by taking the square root : Quadratic functions & equationsVertex form : Quadratic functions & equationsSolving quadratics by factoring : Quadratic functions & equationsThe quadratic formula : Quadratic functions & equations
 Completing the square intro : Quadratic functions & equationsMore on completing the square : Quadratic functions & equationsStrategizing to solve quadratic equations : Quadratic functions & equationsQuadratic standard form : Quadratic functions & equationsFeatures & forms of quadratic functions : Quadratic functions & equationsComparing quadratic functions : Quadratic functions & equationsTransforming quadratic functions : Quadratic functions & equations
 Irrational numbers
 Mastery unavailable
 Irrational numbers : Irrational numbersSums and products of rational and irrational numbers : Irrational numbersProofs concerning irrational numbers : Irrational numbers
 Creativity in algebra
 Mastery unavailable
 Creativity in algebra : Creativity in algebra
 Course challenge
 Test your knowledge of the skills in this course. Have a test coming up? The Course challenge can help you understand what you need to review. Start Course challenge
 Community questions
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Algebra 1 | Math | Khan Academy
https://www.khanacademy.org/math/algebra
https://www.mathplanet.com/education/algebra-1
Algebra 1 – Mathplanet
2018-08-21
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<div><div>
 <main>
 
 
 
 
 <article>
 
	<div>
	<p>Algebra 1 is the second math course in high school and will guide you through among other things expressions, systems of equations, functions, real numbers, inequalities, exponents, polynomials, radical and rational expressions.</p>
<p>This Algebra 1 math course is divided into 12 chapters and each chapter is divided into several lessons. Under each lesson you will find theory, examples and video lessons.</p>
<p>Mathplanet hopes that you will enjoy studying Algebra 1 online with us!</p>
	</div>
 
 </article>
 
	
 
 
 
 </main>
 </div>||||I|||| Mathplanet

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 Algebra 1

 Algebra 1 is the second math course in high school and will guide you through among other things expressions, systems of equations, functions, real numbers, inequalities, exponents, polynomials, radical and rational expressions.

 This Algebra 1 math course is divided into 12 chapters and each chapter is divided into several lessons. Under each lesson you will find theory, examples and video lessons.

 Mathplanet hopes that you will enjoy studying Algebra 1 online with us!

 * Discovering expressions, equations and functions

 * Exploring real numbers

 * How to solve linear equations

 * Visualizing linear functions

 * Formulating linear equations

 * Linear inequalities

 * Systems of linear equations and inequalities

 * Exponents and exponential functions

 * Factoring and polynomials

 * Quadratic equations

 * Radical expressions

 * Rational expressions

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Algebra 1 is the second math course in high school and will guide you through among other things expressions, systems of equations, functions, real numbers, inequalities, exponents, polynomials, radical and rational expressions.
This Algebra 1 math course is divided into 12 chapters and each chapter is divided into several lessons. Under each lesson you will find theory, examples and video lessons.
Mathplanet hopes that you will enjoy studying Algebra 1 online with us!
Algebra 1 – Mathplanet
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 * Course summary
 Polynomial arithmetic
 Intro to polynomials : Polynomial arithmeticAverage rate of change of polynomials : Polynomial arithmeticAdding and subtracting polynomials : Polynomial arithmetic
 Multiplying monomials by polynomials : Polynomial arithmeticMultiplying binomials by polynomials : Polynomial arithmeticSpecial products of polynomials : Polynomial arithmetic
 Complex numbers
 The imaginary unit i : Complex numbersComplex numbers introduction : Complex numbersThe complex plane : Complex numbers
 Adding and subtracting complex numbers : Complex numbersMultiplying complex numbers : Complex numbersQuadratic equations with complex solutions : Complex numbers
 Polynomial factorization
 Factoring monomials : Polynomial factorizationGreatest common factor : Polynomial factorizationTaking common factors : Polynomial factorization
 Factoring higher degree polynomials : Polynomial factorizationFactoring using structure : Polynomial factorizationPolynomial identities : Polynomial factorizationGeometric series : Polynomial factorization
 Polynomial division
 Dividing polynomials by x : Polynomial divisionDividing quadratics by linear factors : Polynomial divisionDividing polynomials by linear factors : Polynomial division
 Polynomial Remainder Theorem : Polynomial division
 Polynomial graphs
 Zeros of polynomials : Polynomial graphsPositive and negative intervals of polynomials : Polynomial graphsEnd behavior of polynomials : Polynomial graphs
 Putting it all together : Polynomial graphs
 Rational exponents and radicals
 Rational exponents : Rational exponents and radicalsProperties of exponents (rational exponents) : Rational exponents and radicalsEvaluating exponents & radicals : Rational exponents and radicals
 Equivalent forms of exponential expressions : Rational exponents and radicalsSolving exponential equations using properties of exponents : Rational exponents and radicals
 Exponential models
 Interpreting the rate of change of exponential models : Exponential modelsConstructing exponential models according to rate of change : Exponential modelsAdvanced interpretation of exponential models : Exponential models
 Logarithms
 Introduction to logarithms : LogarithmsThe constant e and the natural logarithm : LogarithmsProperties of logarithms : Logarithms
 The change of base formula for logarithms : LogarithmsSolving exponential equations with logarithms : LogarithmsSolving exponential models : Logarithms
 Transformations of functions
 Shifting functions : Transformations of functionsReflecting functions : Transformations of functionsSymmetry of functions : Transformations of functionsScaling functions : Transformations of functions
 Putting it all together : Transformations of functionsGraphs of square and cube root functions : Transformations of functionsGraphs of exponential functions : Transformations of functionsGraphs of logarithmic functions : Transformations of functions
 Equations
 Rational equations : EquationsSquare-root equations : EquationsExtraneous solutions : Equations
 Cube-root equations : EquationsQuadratic systems : EquationsSolving equations by graphing : Equations
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 Unit circle introduction : TrigonometryRadians : TrigonometryThe Pythagorean identity : TrigonometryTrigonometric values of special angles : Trigonometry
 Graphs of sin(x), cos(x), and tan(x) : TrigonometryAmplitude, midline and period : TrigonometryTransforming sinusoidal graphs : TrigonometryGraphing sinusoidal functions : TrigonometrySinusoidal models : Trigonometry
 Modeling
 Modeling with function combination : ModelingInterpreting features of functions : ModelingManipulating formulas : Modeling
 Modeling with two variables : ModelingModeling with multiple variables : Modeling
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<div><div><p>Algebra is a term describing any type of math in which letters take the place of numbers in a formula. For example, x + 2 = 5 is an algebraic formula in which x represents the number 3. Taking this concept further allows you to solve many different kinds of formulas, including calculating the area of a shape. It also lets you quickly perform calculations that would otherwise be complex and lengthy. Algebra is useful in everyday life, giving you the foundation to make sound financial and logical decisions. It’s also the building block for every other higher type of mathematics and other subjects. Whether you are calculating distance or deciding how much paint to buy for your walls, algebra will be the tool of choice.</p></div></div>||||I||||          Skip to content
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              Frequently asked questions

                  What is Algebra and why is it important?

                      Algebra is a term describing any type of math in which letters take the place of numbers in a formula. For example, x + 2 = 5 is an algebraic formula in which x represents the number 3. Taking this concept further allows you to solve many different kinds of formulas, including calculating the area of a shape. It also lets you quickly perform calculations that would otherwise be complex and lengthy. Algebra is useful in everyday life, giving you the foundation to make sound financial and logical decisions. It’s also the building block for every other higher type of mathematics and other subjects. Whether you are calculating distance or deciding how much paint to buy for your walls, algebra will be the tool of choice.

                  What careers use Algebra?

                      There are countless careers that rely on algebra. It is absolutely essential for any career as a scientist or engineer, whether it’s chemistry, biology, social sciences, physics, or food science. Computer programming and software engineering, some of the fastest-growing careers today, are heavily reliant on algebra. Algebra is also important for teachers, doctors, and veterinarians. For those interested in technology, algebra is an indispensable tool. New technologies usually have algebra at the base of their functionality. Even careers that don’t directly require using algebra every day will require taking the class in college or high school, meaning it is a prerequisite for those careers. For this reason, it’s safe to assume that algebra is a building block for just about any career out there.

                  What skills should I know before learning algebra?

                      In order to be able to learn algebra, you need to be fluent with basic topics covered in elementary math. This includes arithmetic operations like addition, subtraction, multiplication, and division, as well as an understanding of how fractions work. It is also important to have a solid grasp of non-integer numbers like decimals and percentages, as well as how they’re related to one another. Proportions and ratios are also important in algebra, as is probability. Knowing formulas for basic geometrical shapes also helps: it’s great to enter algebra knowing the formula for the circumference of a circle, the area of a rectangle, perimeters, and other basic formulas. Also, be sure to understand the concept of what an integer is and how the number line works.
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          Algebra Courses
            Learn more about Algebra
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              Frequently asked questions
                  What is Algebra and why is it important?
                      Algebra is a term describing any type of math in which letters take the place of numbers in a formula. For example, x + 2 = 5 is an algebraic formula in which x represents the number 3. Taking this concept further allows you to solve many different kinds of formulas, including calculating the area of a shape. It also lets you quickly perform calculations that would otherwise be complex and lengthy. Algebra is useful in everyday life, giving you the foundation to make sound financial and logical decisions. It’s also the building block for every other higher type of mathematics and other subjects. Whether you are calculating distance or deciding how much paint to buy for your walls, algebra will be the tool of choice.
                  What careers use Algebra?
                      There are countless careers that rely on algebra. It is absolutely essential for any career as a scientist or engineer, whether it’s chemistry, biology, social sciences, physics, or food science. Computer programming and software engineering, some of the fastest-growing careers today, are heavily reliant on algebra. Algebra is also important for teachers, doctors, and veterinarians. For those interested in technology, algebra is an indispensable tool. New technologies usually have algebra at the base of their functionality. Even careers that don’t directly require using algebra every day will require taking the class in college or high school, meaning it is a prerequisite for those careers. For this reason, it’s safe to assume that algebra is a building block for just about any career out there.
                  What skills should I know before learning algebra?
                      In order to be able to learn algebra, you need to be fluent with basic topics covered in elementary math. This includes arithmetic operations like addition, subtraction, multiplication, and division, as well as an understanding of how fractions work. It is also important to have a solid grasp of non-integer numbers like decimals and percentages, as well as how they’re related to one another. Proportions and ratios are also important in algebra, as is probability. Knowing formulas for basic geometrical shapes also helps: it’s great to enter algebra knowing the formula for the circumference of a circle, the area of a rectangle, perimeters, and other basic formulas. Also, be sure to understand the concept of what an integer is and how the number line works.
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Top Algebra Courses Online - Updated [June 2024]
https://www.udemy.com/course/algebra-2-explained-tl/