# Questions tagged [abstract-algebra]

For questions about the study and teaching of abstract algebra, including topics such as groups, rings, fields, and vector spaces.

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### Visual aids for understanding group theory

I want ideas for pictorial representation of groups which can help one understand the different group theorems. Here are some examples of the type of thing I am looking for. In this video by socratica ...
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### Should one teach to use equality or isomorphism in particular groups?

I am wondering the following. Suppose we have some particular space $X$ and $x_1,x_2,x_3,x_4\in X$ has the law of composition that works like Klein four-group. Is it correct to say that the structure ...
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### What is a good way to broach the subject of abstract algebra, to a student in Calc 1, or pre-calc?

Background I want to introduce my students to some big names in mathematics, and one of the names I want to bring up is Richard Borcherds, known for his contributions to the fields of number theory, ...
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### is it beneficial to encourage high school students to conduct their own 'research' in mathematics?

Is it good idea to encourage students to look up more about particular topics that interest them? The idea is that I don't think they will understand how to read math literature. for example, if a ...
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I am a high school student really into algebra and algebraic geometry. I want to expose this sort of math to other high school students that have the motivation and ability. For a long time, I had no ...
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### Can you talk about (the rest of the) field axioms when the operations are not closed? [closed]

Note: Updated based on this. In my course, my instructor posed the following exercise: Let $S$ be the subset of $\mathbb R^n$, $S=\{(a_1,a_2,a_3...a_n) | a_2 = \pm a_1, a_3=...=a_n=0 \}$. Define ...
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When I was an undergrad studying abstract algebra, we used the second edition of Artin and covered groups first and then rings. Fields, vector spaces, and algebras came later, I think. I remember ...
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### in what sense is the subject of finite group theory 'algebraic'?

[cross posted from mse] the class of all finite groups is not closed under produtcs - example: the product over all finite cyclic groups - thus it is not a variety of algebras, ie, it's not ...
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### Rings in parallel with groups in abstract algebra

In a previous question, I asked about the pros and cons of teaching rings before groups in abstract algebra. Recently, it has come to my attention that there is a third approach - a unified approach - ...
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### Improving exposition of a proof about polynomials over infinite fields

This question concerns teaching a proof of the theorem that if a polynomial $f \in k[x]$ over an infinite field $k$ is the zero function (i.e. $f(a) = 0$ for all $a \in k$) then it is also the zero ...
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### Choice of textbook for an undergraduate abstract algebra course

Currently a 5th year PhD student, and I've been fighting tooth and nail to teach one of our junior-year honors sections in undergraduate algebra next fall (desperately hopeful we'll be able to return ...
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### Applications of abstract algebra outside of mathematics and suitable textbook

The question What are some good mathematical applications to present in an abstract algebra course? asks about mathematical applications of abstract algebra. What are some applications of abstract ...
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### How to naturally encounter the properties of identity, commutativity, associativity, and distributivity (to define rings)?

(Cross posted at MSE: https://math.stackexchange.com/questions/3742948/how-did-we-isolate-the-properties-of-identity-commutativity-associativity-and) In elementary school, I remember learning about ...
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### The term "unique" for functions and operations

This is long so... TLDR: Proposing the math community steer away from the misleading term unique, with respect to functions and algebraic operations. Instead, use unambiguous. Why not? Analysis below....
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### Introductory real analysis before or after introductory abstract algebra?

What are the pros and cons for students of taking introductory real analysis before or after introductory abstract algebra, assuming they are going to take both? I recognize that the overlap between ...
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### MacLane-Birkhoff's "Algebra" vs Jacobson's "Basic Algebra I,II" vs Lang's "Algebra"

(Cross-posted at Math.Stackexchange) I'm searching for an apt textbook(s) on Abstract Algebra for a very advanced undergraduate/graduate level course in Algebra, and would be grateful for any help. ...
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### Online open-course-ware that uses Maclane's book "Algebra"

I am struggling with that book which I find to be more of second-guessing type than a book for self-study: it has cryptically written sections, no examples (and those given, and rarely, are even more ...
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### Where do students learn to solve polynomial equations these days?

When I was a math undergraduate 30 years ago in India, we were taught what was then called "classical algebra" (as opposed to abstract algebra), and we were taught among other things solving ...
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### Group theory by geometry

I'm introducing my kids to the concepts of group theory. To make abstract things tangible, I'm trying the geometry way, adopting Arnold's in "Abel's Theorem", so far I've explained, by using symmetry ...
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### Alternating group without $S_n$

I'm going to start introducing my abstract algebra class to a variety of groups soon. Dihedral groups $D_n$ arise out of symmetries on polygons. And the Symmetric group $S_n$ makes sense as the group ...
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### Geometry textbook with an abstract algebra emphasis

I'm teaching a variety of undergraduate and graduate geometry classes (mostly for in-service teachers) which range from elementary axiomatic geometry to more advanced transformational geometry. I'm ...
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### According to Nathan Jacobson, what is Intermediate Algebra and Advanced Algebra?

Nathan Jacobson's Basic Algebra I, II covers many topics in Algebra that is probably even beyond many pure mathematics full professor's scope of knowledge, unless the professor is specialised in ...
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### At what point in the curriculum should the tensor product be introduced?

I remember my linear algebra teacher mentioning tensor products as an advanced topic that would be covered in upper level algebra coursework. During undergraduate abstract algebra, tensor products ...
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### A good example to show group actions and Burnside's lemma

I want to make a presentation of Burnside's lemma outside of group theory, and more as the stand-alone combinatorial tool that it can also be. My plan right now is to make it into a 15-20 minute video,...
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### Open Rings-First Abstract Algebra Text

Building off my own experience and the responses to "Rings before groups in abstract algebra?" I've decided to teach Abstract Algebra using a rings-first approach. However the various texts mentioned ...
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### What made (abstract) algebra grow in relative importance?

Nowadays, when I look at mathematics programs of study, "algebra" (at the abstract level) and "analysis" are treated as equally important. I'm "dating" myself, but this did not appear to be true in ...
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### How to teach abstract algebra for the first time?

I am a Ph.D student in computer science. I am TAing one course this semester, which requires the basics of abstract algebra like rings, fields, ideals, and basic theorems about them. I have done two ...
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### Examples of basic non-commutative rings

I am teaching an intro to ring theory, and after grading the first quiz, I realize most of my students are under the assumption that rings must be commutative. I have given them the example of ...
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### Looking for online abstract algebra courses making use of computer algebra systems

preferably areas of algebra of value and interest to computing practitioners. any level from introductory to (say) Grobner bases. preferably using open source computer algebra software. no preference ...
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### Explaining genus to students

I need to do a presentation on my thesis, which is in arithmetic geometry. This presentation is meant for all students of mathematics, but I will assume some knowledge of abstract algebra (i.e. groups,...
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### Is MacLane and Birkoff's "Algebra" suitable today as either an undergraduate or graduate text in abstract algebra?

I'm going to soon review the 3rd edition of Saunders MacLane And Garrett Birkoff's Algebra at my blog soon and this is the first time I'm really carefully reading it. While I'm really enjoying the ...
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### Content for a two-quarter class on abstract algebra

I am going to teach a two-quarter sequence on abstract algebra at a mid-size american public university. Ideally, this course would introduce groups, rings, and fields, then end with some ...
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### Simple examples that violate group axioms

In a course for non-math-majors at a liberal arts college, I would like to give a few lectures and activities about groups and symmetry. I think it's straightforward to explain the group axioms and ...
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### What are the "best" groups to use as examples while learning new concepts in algebra?

This question was asked to Math SE at first but it seems like it is more appropriate to ask it here. While learning new concepts in algebra it is quite helpful to check some examples which includes ...
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### Visual representation of Cartesian Product of groups

I'm today trying to construct a lesson on theory and problems of group theory. A specification of this lesson would be to use as much visual representations as possible. I have found some trivial ways ...
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### How can one convincingly present the alternating group?

I am soon going to explain to my students what the alternating group is. The definition is subtle.... one must prove that the notation of an "even permutation" is well defined. There seem to be two ...
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### Group theory via actions

I'm teaching undergraduate group theory [again] this term. I've been increasingly dissatisfied with the approach in various books (I've used Fraleigh and looked at others). They're all reasonable ...
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### What toys and props illustrate concepts from abstract algebra?

I am looking for toys or props that illustrate concepts from abstract algebra. The best known example is of course the Rubik's Cube; another great example is the Fifteen Puzzle, especially if like Sam ...
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