# Tag Info

### 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 ...

### 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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### 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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### 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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### 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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### Simple examples that violate group axioms

I can't believe no one has mentioned (positive, say) integers under exponentiation as a non-associative operation. It's by far my favorite, though of course this example is also not commutative and ...
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### Simple examples that violate group axioms

For something that's not commutative, try rotations about axes through the origin in 3D. A simple rectangular book can illustrate this very well. Try rotating the book in 90 degrees around vertical ...
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### What are the "best" groups to use as examples while learning new concepts in algebra?

You mentioned the linear groups over the real numbers, which I think is a useful example, with the potential drawback that it is infinite. To "remedy" this one can consider the linear groups over ...
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### Where do students learn to solve polynomial equations these days?

In the United States, solving linear and quadratic equations is a standard part of Algebra 1, which most students take in 8th or 9th grade. Students will return to polynomials and see long division ...

### in what sense is the subject of finite group theory 'algebraic'?

Typically, students learn about finite groups long before they think of "variety of algebras" or "compactness theorem". I suppose you also want to exclude fields from "...
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### Third isomorphism theorem: how important is it to state the relationship between subgroups?

I agree that $\text“\color{red}{N \text{ is a normal subgroup of a group }H}\text”$ is a redundant/bogus condition of the given theorem. To specify what is rightly a $\color{red}{\textrm{lemma}}$ of ...
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### Introduction of group action as morphism of groups

I prefer just to keep the language as simple as possible and the level of abstraction as low as the circumstances allow when teaching courses for students who see the material for the first time. So ...
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### What are the "best" groups to use as examples while learning new concepts in algebra?

In addition to the dihedral groups, the quaternion group $Q_8$, the symmetric and alternating groups, and small matrix groups, I have always found the following examples useful: Automorphism groups ...
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### Content for a two-quarter class on abstract algebra

Abelian group theory emphasis with application being point symmetry (playing cards, stars, polyhedra, buildings, molecules, etc.. I wouldn't go into IR stretching modes, but you could at least ...
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### Looking for online abstract algebra courses making use of computer algebra systems

There's Abstract Algebra: Theory and Applications by Judson and Beezer. It doesn't get to Grobner bases but does cover the standard material (and some non-standard stuff, too). It has Sage code and ...
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### Explaining genus to students

As a fellow arithmetic geometer working on some problems related to Faltings' theorem, I've given a fair number of explanations of algebraic curves to non-specialists, so I can tell you how I usually ...
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