I already wrote about Set theory and I eventually want to write about symmetry, so this seems like the perfect opportunity to discuss Group theory. Let’s start where we left off with Set theory. We had a very abstract concept of a Set: a collection of “things”. Then we added some basic structure to that collection (the operation of set concatenation), and before we knew it we were counting and adding.
Well, to formalize it a bit, when you take a set and add a binary operation, and an identity element, then what you get is called a monoid. Monoids are extremely abstract yet very powerful. (In a way they even form the underpinning of all of computer science, where the set is a collection of functions, and the product is simply chaining these functions together, called function composition, and the identity is simply a “noop“.) I’ll talk more about this in a future post because it is so cool how these ideas tie together.
A Group is just like a Monoid, except every element in the set must also contain its inverse. When an element operates on its inverse, it results in the identity! Some basic examples, are the integers, , where every element also has its addtive inverse, called its “negative”. Another example are the rationals, , where every element has its multiplicative inverse, called its “reciprocal”. In the former example, the operation is addition and addtive identity is 0, while in the latter the product is multiplication and the multiplicative identity is 1. Both of these groups also commute (i.e. the order of operation does not matter), and we call these groups “abelian“. They are named after the mathematician Niels Henrik Abel, but strangely the word is not capitalized.
These groups may seem pretty abstract at first, but here are a few more tangible examples. First, is that to physicists, a set with elements and their inverses will immediately remind one of a set of particles and their antiparticles. But we can actually get much more tangible than that if we pull the same trick we pulled above with our monoid: instead of thinking about objects, get Categorical, and think about processes. It’s easy to imagine a set of transformations and their inverses: raise and lower, shrink and grow, cut and paste. All of these examples have the identity of “do nothing”.
Groups are strongly related to symmetry because an object which is equivalent to the original after transformations is at the heart of symmetry! In future posts we will talk about the groups of rotations and reflections to explore these concepts.