29 votes
Accepted

How should normal subgroups be introduced?

First of all, I should point out that the standard definition of a normal subgroup is A subgroup $N \subset G$ is normal iff $g n g^{-1} \in N$ for all $n\in N$ and $g\in G$. When I say "the" ...
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  • 7,930
26 votes

Rings before groups in abstract algebra?

My favorite textbook for an undergraduate course in Abstract Algebra, Ted Shifrin's Abstract Algebra: A Geometric Approach, uses a rings-first approach. The primary pro is that students are much more ...
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24 votes
Accepted

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

It is possible to usefully mention "Lie groups (and Lie algebras)" in an introductory course, if one does not give formal definitions, but, rather, examples. It is not necessary (or advisable) to "...
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  • 13.4k
21 votes
Accepted

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

$\mathbb Z_n$ since it is very easy to compute in and you have one of order $n$ for every $n\ge 1$. $S_3$, since it's the smallest non-abelian group. $S_4$, since $S_3$ is sometimes too small. $...
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  • 2,941
17 votes

How should normal subgroups be introduced?

The way I like to approach this is as follows. After discussing subgroups, the natural question as to forming the quotient $G/H$ arises. I then proceed to look at the cosets and prove that if $gH\cdot ...
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  • 2,941
17 votes

What makes cosets hard to understand?

I strongly suspect the difficulty is not with cosets specifically, but with working with equivalence relations generally, especially when combined with objects that they have only recently become ...
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16 votes

Rings before groups in abstract algebra?

I have taught both groups first and a rings first course. When I was a post-doc at Rutgers University, I taught their standard introduction to modern algebra course using Hungerford's undergraduate ...
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  • 607
16 votes

Simple examples that violate group axioms

Combining colored paint is an interesting example of a non-associative operation. Define $Paint_1 * Paint_2$ to be the paint obtained by mixing the two paints in a $1:1$ ratio. It is easy to see that ...
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  • 1,426
16 votes

Is MacLane and Birkoff's "Algebra" suitable today as either an undergraduate or graduate text in abstract algebra?

Preface. Birkhoff & Mac Lane's Algebra is a brilliant book. I should probably spend some time with it again, actually. Also, I apologize for such a long response. I think too much about algebra ...
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15 votes
Accepted

Graphing functions from a finite field to itself

Just supplementing Benjamin Dickman's nice answer, here is $x \mapsto x^2 - x$ in $\mathbb{Z}_{18}$ in the same style: For example, the pentagon wheel reflects the fact that $$(5+3k)^2-(5+3k) = 9k^2 ...
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15 votes

Examples of basic non-commutative rings

The quaternion ring is a pretty simple example of a non-commutative ring (a skew-field, even).
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14 votes
Accepted

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

When I introduce groups I first go over a very (e.g., 15 examples) long list of particular groups. I go over each example and verify the group axioms without naming it such. Then I present the problem ...
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  • 2,941
14 votes
Accepted

Impossibility of trisecting the angle, doubling the cube and alike, what are reasons for or against discussing them in a course on algebra?

Let me first say how I have taught this, and then why it was worth doing. Here is the stripped down version I speak of. The first block of bullet points is one day. Forget about straight edge and ...
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14 votes

What are the "best" groups to use as examples while learning new concepts in algebra?

Symmetry groups of basic regular polygons and polyhedra. For one thing, they are noncommutative; for another, sometimes they coincide with other "known" groups; for a third, there can be physical ...
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  • 5,752
13 votes

Graphing functions from a finite field to itself

One of the approaches taken in some areas of mathematics (e.g., in arithmetic dynamics and considerations of preperiodic points, etc) is to create these graphs by drawing discrete points and then ...
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13 votes

The term "unique" for functions and operations

I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way. Students have trouble with the notion of a function because it's hard. The way ...
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12 votes
Accepted

Can GAP/Magma be helpful for a first abstract algebra course?

I TA'd a first course in abstract algebra during my senior year of undergrad. The professor wanted a computational flavor to the course, so we introduced Magma right off the bat. We wanted to allow ...
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12 votes
Accepted

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

I think you are correct to worry about an overload of algebraic structures, especially if they are not well-motivated. I would strongly encourage you to keep the naming of various algebraic ...
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12 votes

Rings before groups in abstract algebra?

I've done it both ways, although I do rings-first now and for the foreseeable future. I think the pros and cons have a lot to do with the audience, especially if there are a lot of pre-service ...
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11 votes

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

I suggest having a look at the following book by John Stillwell: Naive Lie Theory. A fast-paced week or (if you're lucky enough... or believe it's worth it as I would) two weeks could be created by ...
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11 votes
Accepted

How to teach abstract algebra for the first time?

I second Adam. Keep it with strong with examples if you can. There is a mathoverflow answer with a ton of group theory examples. You might want to force them to practice doing more "rote" exercises ...
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  • 2,584
11 votes

Where do students learn to solve polynomial equations these days?

Students learn linear and quadratic equations in high school algebra. And then, if they have forgotten it, re-learn it in college, in courses called "pre-calculus" or something. Unless they ...
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  • 6,171
10 votes

Free Groups First Approach

Describing groups by generators and relations (i.e., as quotients of free groups) is not a good way to describe them most of the time. There are some exceptions, Coxeter groups, braid groups, but ...
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  • 13.4k
10 votes
Accepted

How can one convincingly present the alternating group?

It seems to me that the first thing to do is to define the signature. It is a very important morphism, and you need to have a bunch of example of morphisms to present to your students anyway. Only ...
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10 votes

Impossibility of trisecting the angle, doubling the cube and alike, what are reasons for or against discussing them in a course on algebra?

Perhaps rather than spend time establishing that trisecting an angle is impossible via Euclidean (ruler-compass) constructions, you could instead (a) Make that claim without proof, and (b) ...
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10 votes

Simple examples that violate group axioms

I really like the example of the game Rock-Paper-Scissors, thinking of it as a binary operation $\star$ on the set {Rock, Paper, Scissors}. The rules are: Rock$\star$Paper $\mapsto$ Paper Rock$\...
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  • 7,686
10 votes

Examples of basic non-commutative rings

Why not turn $M_2(\mathbb{R})$ into a multiplication on $\mathbb{R}^4$? Here's a fun, somewhat weird, ring: $$ \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \left[ \begin{array}{...
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9 votes

What are some good mathematical applications to present in an abstract algebra course?

Burnside's lemma can be covered in one day (or less) in an abstract algebra course, and has lots of wonderful applications to counting problems.
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  • 7,930
9 votes

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

I think in a first abstract algebra course the goal is simply to make students aware that such things exist, give a couple of examples, and let them know that there is much, much more that can be ...
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  • 16.1k
9 votes

At what point in the curriculum should the tensor product be introduced?

(This pertains to U.S. universities.) Times may have changed, but when I was in graduate school (several places, and yes I know this is unusual, but I mention it because I'm talking about more than ...
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