tl;dr lectures please

The Socratica abstract algebra videos was my introduction to advanced math and the subject of abstract algebra is still a favourite of mine. My gratitude to the Socratica team for first introducing these magical mathematical objects of all kinds into my life.

How to use this is not the material to which you can doze off for even a second. Full concentration and rewind freely. Consider it a challenge, because it is and it feels like it. Like riding a dragon. (yes)

what is it

The fundamental object in abstract algebra is the group construct.

A group consists of the very basics needed to construct an algebra (!), including of course but not limited to, the one you know. (well hello mr construct your own algebra)

A group is a set of elements for which it holds that

  • it has an operation that associates an element to every pair of elements (closed under the operation)
  • it is associative (no parenthesis needed)
  • there is an identity element
  • every element has an inverse (the two combined equals the identity)

The basic group object with additional properties then goes on to define rings, fields, vector spaces etc.

This is specifically what it takes to build an algebra. And we’ve only recently found this out, its a new field.

Really grokking this, group theory, its hard to describe but I feel it has changed me, upgraded my appreciation of math and life in general.

Its just beautiful and very elegantly simple in its own right, it becomes a reason to start learning proof writing* and just feels completely like working with Matrix (the movie) code.

Also personally, I like the name abstract algebra, so mysterious.** Other equally valid names are modern algebra (boring) and just algebra (among mathematicians).

the book

The book I use to learn this stuff is A Book of Abstract Algebra. This particular book just so happens to be my favourite book of all time (for some years). Just a masterpiece.

Do not (under any curcumstances) skip the exercises (ever in math literature) or you will not be doing math at all plus a significant part of the material is found only in the exercises.

I feel there is a clear demarcation in my life: before and after I wrote out all the possible operations on a two-element group, an exercise in the beginning of the book. These exercises feels like really doing math.

the proofs book

The proof writing book I used is Chartrand’s Mathematical Proofs: A Transition to Advanced Mathematics and I loove it. Very highly recommended, if expensive. I bought a used second version and it did the trick very nicely.

Do not deny yourself the deep, deep pleasure of learning how to write proofs, however simple. It starts with proving even plus even equals even, a gentle start but also a full basejump as the joy of actually writing proofs feels completely independent of the complexity of the proof. So exciting!

*) Spoiler alert, steals the show… Marks the spot where math starts.

**) And it namedrops like a cartoon elephant from the sky with a little hat on.