# Tag Info

28

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" standard definition, I mean that this is how working group theorists think of normal subgroups, and this is one of two basic ways to prove that a subgroup is normal....

24

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 "define" smooth manifolds, which seems to have considerable baggage-of-abstraction of its own. Just give important examples, noting that they do seem to have a lot ...

24

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 familiar with examples of rings (integers, polynomials) than they are with the standard examples of groups (symmetries of simple shapes, permutations). Indeed,...

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$\mathbb Z_n$ 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. $A_4$, Since it is the smallest group that does not contain a subgroup of every possible order dividing its order. $\mathbb Z_2 \times \mathbb Z_2$ since it's ...

16

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 $gH\cdot g'H=gg'H$ is a well-defined operation, then the cosets become a group, which we call the quotient group. This is a very easy proof with basically nothing to do....

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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 acquainted with. So apologies in advance that this answer deals more generally with equivalence relations and my opinion that they often need more coverage than ...

16

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 $$(Red * Blue) * White \neq Red * (Blue * White).$$ They are different shades of purple. I haven't tried it, but it should be easy enough to make a ...

16

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 pedagogy and textbooks. The short version is I think the book can be used for either undergraduates or graduates with some success, but I think it is less than ...

15

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 of studying all of these examples together, motivated by simply saving labour. Instead of proving things again and again for each case, we'd like a single ...

14

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 $$(5+3k)^2-(5+3k) = 9k^2 + 27k + 20 = 9k(k+3) + 20 = 2\bmod 18 \;.$$

14

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 algebra text. I was kind of annoyed at the time that I would (unless I wanted to fight the textbook) have to teach using a rings first approach. By the end of ...

14

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

13

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 meaning to some concepts in group theory. And they are very "hands-on" - it can be fun to make sure students understand how to generate the symmetry group for the ...

13

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 they're going to get a handle on it is by struggling with it, encountering the hard parts of the definition, and finding and eliminating their misconceptions ...

12

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 the students to experiment with permutations without needing to actually do large computations themselves. As part of my contribution to the course, I was put ...

12

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 using arrows to show which values map to which other ones. Figuring out a "canonical" way to draw these pictures might be a bit tough (this is related in some ...

12

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 structures to a minimum in a first course in abstract algebra. In my view, the goal of the course is for them to see the power of abstraction in a fairly concrete ...

11

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 pulling the big ideas from chapters 1, 2, and 4. I would argue that this shouldn't be the first time the students are seeing groups like SO, SU, etc. Those ...

11

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 mathematics teachers taking the course (like there are at my place). The main "pro" is pedagogical: Rings are more familiar objects to students and, for pre service ...

11

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 specialize in mathematics at the college level, they do not learn any more. Why not? Because we have computers now, so most people do not need to solve polynomial ...

10

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 then is the alternating group really relevant, and you get all properties you want from what you will have done with the signature. Of course, I am telling you to ...

10

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$\star$Scissors $\mapsto$ Rock Paper$\star$Scissors $\mapsto$ Scissors The operation is certainly commutative, but it is not associative. For example, $$(Rock \... 9 Describing groups by generators and relations (i.e., as quotients of free groups) is not a good way to describe them most of the time. There are some exceptions, Coxeter groups, braid groups, but mostly groups do not arise in that fashion. Further, "the word problem" is undecideable, so it's not the case that there are algorithms (much less efficient ones) ... 9 I think in a first abstract algebra course the goal is simply to make students aware that such things exist, give a couple of examples, and let them know that there is much, much more that can be learned in future classes. With that in mind: Start with SO(2). Note that all elements of SO(2) can be represented in terms of a continuously-varying ... 9 Perhaps rather than spend time establishing that trisecting an angle is impossible via Euclidean (ruler-compass) constructions, you could instead (a) Make that claim without proof, and (b) Mention that different axioms do permit angle trisection: (Figure from Geometric Folding Algorithms: ... 9 Let me first say how I have taught this, and then why it was worth doing. Here is the stripped down version I speak of. The first block of bullet points is one day. Forget about straight edge and compass, no one uses them. Fortunately, I think we still have another few years of students who have still seen the standard five function calculator: +, -, \... 9 Why not turn M_2(\mathbb{R}) into a multiplication on \mathbb{R}^4? Here's a fun, somewhat weird, ring:$$ \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \left[ \begin{array}{cc} x & y \\ z & w \end{array}\right] = \left[ \begin{array}{cc} ax+bz & ay+bw \\ cx+dz & cy+dw \end{array}\right]$$hence$$ (a,b,c,d) \star (...

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(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 one data point) this was a standard topic that showed up somewhere in the standard first year (2-semester) graduate algebra sequence (e.g. Lang, Hungerford, ...

9

The rational root theorem, synthetic division, the remainder theorem, Descartes rule of signs, and similar lower level topics were fairly widely taught in U.S. high school algebra-2 courses before and during the 1980s, but they've slowly been de-emphasized as graphing calculators (allows for numerical equation solving) came onto the scene at the end of the ...

9

Despite the names of these fields, as a student I found real analysis more abstract than abstract algebra: real analysis was less real and more abstract to me than abstract algebra. I don't think I can justify this, but let me give two examples: Lagrange's theorem in abstract algebra: The order of a subgroup $H$ of a finite group $G$ divides the order of \$...

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