﻿On this day, I explained to Adithya, or Aleph0, some math including group theory.


While explaining group theory, I finally got group theory to click.


Group theory is the study of groups. And a group is just a set with structure, namely the structure of an operator x that takes in two elements of the group, and returns a third, with the extra requirements that the map is associative, has a neutral element, and that each element has an inverse (which means that when an element and its inverse are put together, you get the neutral element).


Coming up with a group seemed easy. You just do the same thing you did with fields, or vector spaces, or anything set+structure category combo.


The problem is 3blue1brown. In this introduction to group theory, he talks about how the rotations of a square that leave it being the same are a group. To me I’m like WTF, that feels a lot more category theoretic to me. And this approach uses a much more “symmetric” approach to group theory, basically saying that any symmetry you spot in the world is somehow a group.


Well that’s the problem. It’s not immediately obvious what the group is. What is the set? And what is the operator?


Well talking to Aleph0, I realized that the homomorphisms are elements of the set, and that composition is the operator. In hindsight this feels so obvious. What does group theory have to do with symmetry? Technically nothing. Only that one of the main groups that you play around with are groups created from a category.


Basically what you do is you look at some category. You identify what a structure preserving map means in that category. You then need a specific example of the category (for example with vector spaces you can give the classic R^3 vector space over R. Or with sets you can give the set {1,2,3,4}). You finally ask yourself the question: consider the set of all structures preserving maps from this thing to itself. This set, which is a set of maps, is a group when equipped with map composition.


So symmetry really just means: turning a category example into a group.


This approach clarifies so many things. For example, consider the set {1,2,3,4}. The symmetries of that group are S_4, which is used everywhere in group theory. It’s just the set of all homomorphisms on the set to itself, which is essentially all the bijections from the set to itself, which is essentially all the possible combinations you can assign the numbers in the first set to the second. Obviously this group is important because it feels very fundamental.


But again the point is whenever you have an object in a category, you can form a group from it by invoking it’s “symmetries”. Quite nice if you ask me. Also quite confusing and I bet very few people understand this.
It also makes understanding group theory much easier because you can now consider that there are only two kinds of groups you really deal with concretely. You have constructive groups, or groups that you built up yourself, and you have “symmetry” groups, or groups ripped of another category, with composition as the operator.