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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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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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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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• 2,941

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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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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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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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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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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