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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Rings before groups in abstract algebra?

The default approach to teaching abstract algebra seems to be groups first, then rings. However, occasionally a textbook pops up (e.g. Childs' A Concrete Introduction to Higher Algebra or Hodge et al'...
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28 votes
5 answers
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How should normal subgroups be introduced?

One standard definition of a normal subgroup is A subgroup $N \subset G$ is normal iff the set of left cosets $\{gN\}$ and right cosets $\{Ng\}$ coincide. There's a class of similar definitions (...
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22 votes
3 answers
974 views

What can be said about Lie groups in a first abstract algebra course?

Lie groups are among the most important examples of groups in mathematics and physics, but they are rarely discussed in introductory undergraduate abstract algebra courses, which tend to focus on ...
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20 votes
10 answers
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What makes cosets hard to understand?

I have been teaching introductory group theory to undergraduates. We reached cosets several weeks ago, but the combination of the textbook, my explanations and various practice questions has left the ...
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20 votes
2 answers
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Impossibility of trisecting the angle, doubling the cube and alike, what are reasons for or against discussing them in a course on algebra?

When I taught courses on algebra giving a first exposition to Galois theory I usually included some discussion of classical results showing the impossibility of constructing certain points with ruler ...
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19 votes
8 answers
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What are some good mathematical applications to present in an abstract algebra course?

One of the main difficulties for a student learning abstract algebra is understanding the motivations behind concepts like groups, normal subgroups, rings , ideals etc. Also, many have difficulty ...
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17 votes
2 answers
731 views

What examples of groups should students in abstract algebra learn to test ideas on?

Having seen many people go through abstract algebra courses, I've noticed that one factor for success is the ability to pull out examples to test ideas on (e.g. if two subgroups have the same index, ...
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16 votes
8 answers
1k views

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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16 votes
4 answers
639 views

What are some good ways to motivate and introduce reasoning abstractly about abstract algebra?

I've found one of the hardest topics to introduce to students early on is abstract algebra. Even if they've already written proofs, it's hard for them to work directly from axioms. They seem to have ...
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16 votes
1 answer
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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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15 votes
7 answers
3k views

What are the differences between popular undergraduate abstract algebra books?

I will be teaching a year-long undergraduate introduction to abstract algebra in the fall, and I am quite looking forward to it! I need to choose a textbook, and I don't have personal experience with ...
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14 votes
6 answers
310 views

Free Groups First Approach

In an undergraduate Abstract Algebra class, it appears that there are two standard approaches: "groups first" or "rings first". Most abstract algebra texts with a "groups first" approach start with ...
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14 votes
3 answers
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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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14 votes
2 answers
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Can GAP/Magma be helpful for a first abstract algebra course?

Does anyone have any experience using either GAP or Magma in a first abstract algebra course? The context here is a course for sophomore/junior math majors that focuses primarily on groups. I'm ...
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13 votes
2 answers
2k views

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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12 votes
6 answers
828 views

Graphing functions from a finite field to itself

I have been teaching a ring theory course this semester, focusing on modular arithmetic and quotient rings of polynomials over fields. Several students have asked me how one could graph functions ...
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12 votes
3 answers
2k views

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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11 votes
9 answers
550 views

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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11 votes
3 answers
377 views

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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11 votes
2 answers
291 views

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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  • 6,350
11 votes
1 answer
402 views

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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  • 219
10 votes
7 answers
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Galois Theory: necessary?

I noticed the discussion of whether the teaching of Galois Theory is necessary on MathOverflow. Here at LSE, everything we teach in mathematics should have some application to the social side of life. ...
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9 votes
3 answers
1k views

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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8 votes
2 answers
213 views

When to cover other algebraic structures in an abstract algebra course?

I am preparing to give an abstract algebra course, which should be mainly focused on Group Theory. However, I want to cover other algebraic structures (magmas, quasigroups, etc.) as well. I am ...
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7 votes
3 answers
306 views

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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7 votes
3 answers
157 views

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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7 votes
1 answer
128 views

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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7 votes
1 answer
209 views

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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  • 173
7 votes
3 answers
434 views

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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  • 171
6 votes
4 answers
347 views

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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6 votes
4 answers
254 views

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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6 votes
2 answers
233 views

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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6 votes
1 answer
611 views

What are some good motivating questions for introductory abstract algebra? [duplicate]

When seeing groups and such for the first time, the abstraction often seems pointless and unnecessary to students. (Most students at my school leave their introductory abstract algebra class thinking ...
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6 votes
3 answers
259 views

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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6 votes
1 answer
544 views

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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5 votes
5 answers
4k views

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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5 votes
3 answers
610 views

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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5 votes
2 answers
486 views

Undergraduate-level abstract algebra books or courses that don't start with groups or rings

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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5 votes
5 answers
330 views

Notation for an element in a polynomial ring

Let $F$ be a field. What is the best notation (in an undergraduate or graduate abstract algebra class) for a generic element of the univariate polynomial ring $F[x]$? The most common notation seems ...
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5 votes
1 answer
193 views

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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  • 6,350
5 votes
1 answer
694 views

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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4 votes
6 answers
442 views

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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4 votes
3 answers
325 views

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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4 votes
1 answer
290 views

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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4 votes
0 answers
249 views

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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  • 4,015
3 votes
3 answers
876 views

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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  • 155
3 votes
4 answers
237 views

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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  • 249
3 votes
2 answers
147 views

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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  • 361
3 votes
2 answers
436 views

Seeking your advice on books for abstract algebra and linear algebra

I am a college sophomore in the US with a major in mathematics and am an aspiring mathematician in the fields of computational complexity theory and cryptography. I would like to seek your advice and ...
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3 votes
2 answers
297 views

Succinct description of situations where naively obvious is correct, but for far more complicated reasons?

What is the name for a situation where the obvious thing turns out to be true, but the reasoning is more complicated? In abstract algebra we can say the rational numbers - the fractions, $\mathbb{Q}...
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