You could forgive mathematicians for being drawn to the monster group, an algebraic object so enormous and mysterious that it took them nearly a decade to prove it exists. Now, 30 years later, string theorists — physicists studying how all fundamental forces and particles might be explained by tiny strings vibrating in hidden dimensions — are looking to connect the monster to their physical questions. What is it about this collection of more than 1053 elements that excites both mathematicians and physicists? The study of algebraic groups like the monster helps make sense of the mathematical structures of symmetries, and hidden symmetries offer clues for building new physical theories. Group theory in many ways epitomizes mathematical abstraction, yet it underlies some of our most familiar mathematical experiences. Let’s explore the basics of symmetries and the algebra that illuminates their structure.
We are fond of saying things are symmetric, but what does that really mean? Intuitively we have a sense of symmetry as a kind of mirroring. Suppose we draw a vertical line through the middle of a square.