Group Theory Overview

A group is an algebraic object that measures symmetry.

There is an abstract and a "dynamic" perspective on groups. We need both, and it is useful to play the two off of each other.

A group is rigorously defined as a set $G$ together with a binary multiplication $G\times G \to G$ satisfying the axioms:

• associativity: $(ab)c = a(bc)$
• identity: there's some element $e$ such that $ex = x = xe$ for all $x\in G$
• invertibility: for every $g\in G,$ there's some $h\in G$ such that $gh = e = hg$ We will see more of this later.

Groups were originally invented to understand symmetries of polynomials. For example, in the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ we can take either plus or minus the square root, and neither one is "better" than the other. The original application of this was to show that the quintic equation $ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0$ does not have a general solution by radicals.

Groups are ubiquitous throughout mathematics.

There is a dichotomy between the study of finite groups and the study of infinite groups. In the latter case, one must usually use geometric tools to understand the groups in question.