I covered this material in a two-semester graduate course in abstract
algebra in 2004-05, rethinking the material from scratch, ignoring
traditional prejudices.
I wrote proofs which are natural outcomes of the viewpoint. A viewpoint is
good if taking it up means that there is less to remember. Robustness, as
opposed to fragility, is a desirable feature of an argument.
It is burdensome to be clever.
The worked examples are meant to be model solutions for many of the standard traditional exercises. I no longer believe that everyone is obliged to redo everything themselves. Hopefully it is possible to learn from others’ efforts.
June, 2007, Minneapolis
The examples are given to assist, not necessarily challenge. The point is not whether or not the reader can do the problems on their own, since all of these are at least fifty years old, but, rather, whether the viewpoint is assimilated. In particular, it often happens that a logically correct solution is conceptually regressive, and should not be considered satisfactory.
Again, algebra is not a unified or linearly ordered body of knowledge, but only a rough naming convention for an ill-defined and highly variegated landscape of ideas.
After proving Lagrange’s theorem and the Sylow theorem, the pure theory of finite groups is not especially emphasized. After all, the Sylow theorem is not interesting because it allows classification of groups of small order, but because its proof illustrates group actions on sets, a ubiquitous mechanism in mathematics.
A strong and recurring theme is the characterization of objects by (universal) mapping properties, rather than by goofy constructions. Nevertheless, formal category theory does not appear.