Introduction

These notes provide an overview of abstract algebra — the study of algebraic structures. That's a big word. What's an algebraic structure? We'll start with a naive answer just to position ourselves. In grade school, we learned a variety of facts:

1+1=22×2=43−2=59÷3=3 1 + 1 = 2 \\ 2 \times 2 = 4 \\ 3 - 2 = 5 \\ 9 \div 3 = 3

Later, we learned that:

−1→can’t do that \sqrt{-1} \to \text{can't do that}

and then later:

−1→i \sqrt{-1} \to i

Most of us know that:

some real number0→undefined \dfrac{\text{some real number}}{0} \to \text{undefined}

But, there's nothing physically stopping us from giving this a symbol:

some real number0:=♡ \dfrac{\text{some real number}}{0} := \hearts

and defining rules on what we can do with that symbol.

♡♡=1♡⋅0=0 \dfrac{\hearts}{\hearts} = 1 \\[1em] {\hearts} \cdot 0 = 0

Now, just because we can doesn't mean it's a good idea. On the one hand, it can make some operations interesting:

1x−x=♡ \dfrac{1}{x - x} = \hearts

However, the longer we play with our new friend, the more we realize how disturbing it is:

1x−x=♡1=♡(x−x)1=♡⋅01=0 \begin{aligned} \dfrac{1}{x - x} &= \hearts \\[1em] 1 &= \hearts(x-x) \\[1em] 1 &= \hearts \cdot 0 \\[1em] 1 &= 0 \end{aligned}

So now we have to decide: Do we want to stay friends with ♡?{\hearts?} Should we abandon our other friend x⋅0=0?{x \cdot 0 = 0?} This a small hint at what abstract algebra deals with. We have a box of objects, rules, and axioms. Abstract algebra looks inside this box and asks questions like:

  1. Are the rules or axioms consistent?
  2. Are there limitations to the rules or axioms?
  3. Can we apply this rule or axiom from this other box?
  4. Is there some structure, or organization, to the things in this box?
  5. ...

As we can likely tell, abstract algebra has a high level of generality. The notes in this volume provide a broad overview of abstract algebra.