The purpose of this post is to serve as a reference I can link to. I define groups, sure, but the goal is really to communicate what they feel like, how I think about them, what I think is neat about them.

How do I describe groups to people?

Many things in life, in art or culture, and in thought have symmetries. Lately I like to say that for a mathematician, a group is the abstraction of the idea of symmetry.

For example, maybe you’ve played with a Rubik’s cube. The different ways of rearranging it are a kind of symmetry. For a mathematician, there’s a group there, which comprises all the ways you can rearrange the Rubik’s cube. This is a finite group, since the cube has six faces with nine squares each, so has at most $(9 \cdot 6)!$ possible states (and in fact far fewer). Infinite groups are possible too.

Now, for a mathematician, the group associated to the Rubik’s cube has a life completely divorced from the cube. I often think of groups as little ghostly abstract gadgets. This abstraction is useful: by taking the symmetries away from the object, you have the ability to make connections between distinct objects if they have the same group of symmetries.

How do I describe geometric group theory to people?

My PhD work is in a field of math called geometric group theory. “Group theory” by itself is the study of groups, so what makes it “geometric” is studying groups from a lens of geometry. One way to think about that is to think about it as realizing “hey wait, the Rubik’s cube can tell us about the Rubik’s group!” If that sounds obvious, you might be a geometric group theorist without knowing it!

What happens a lot in math, I think, is that there is so much of it that we forget to treat all of it with the attention it might deserve. Which is fine! Being able to put aside what isn’t at focus is a useful skill. For mathematicians not focused on symmetry, often a group shows up just as part of the “metadata” associated to some other mathematical object whose interest is more primary.

Have I tried other ways?

Yes! I used to say that groups are “a place where you can play the game of multiplication”. Here’s what I mean. Groups have an operation as part of them. For the Rubik’s group, this operation is “composition” in the sense of “do this, then do that”. I’ll say what this operation must be like in the next section, but one thing to note is that other notions of operation are possible.

For example the (nonzero) real (or complex) numbers form a group where the operation is multiplication. When mathematicians want to highlight the operation in a group, like say you have a group named $G$ and elements of it $g$ and $h$, they use the same notation that you might for multiplication in algebra: so $gh^2$ or $g\cdot h^2$ or $g \cdot h \cdot h$, in the same way you would write $6x^2$ or $6 \cdot x^2$ or $6 \cdot x \cdot x$.

I no longer lead with this description because it hasn’t worked so well for me. Multiplication is already not something people think of themselves as doing frequently, and “playing the game of” leans on the somewhat common mathematical phrase “play[ing] the same game”, used to mean “apply this line of reasoning to a slightly different situation”, which I don’t ever hear outside of mathematical contexts.

What are they?

Formally a group $G$ is a set (so a collection of “elements” which are members of the set with no repetitions) together with a binary operation $\cdot$ that satisfies the following restrictions:

• $G$ is closed under the operation; that is, the operation $\cdot$ defines a function $(\cdot) \colon G \times G \to G$, sending a pair of elements, say $g$ and $h$, to a new element $g\cdot h$. In the analogy with symmetries and composition, this says that if you first transform a symmetric object according to one symmetry and then another, what you have recognized is another symmetry of the object.
• The operation is associative so that $(g \cdot h) \cdot k = g \cdot (h \cdot k)$. The operation need not be commutative: like with matrices in linear algebra, $g\cdot h$ might be very different from $h \cdot g$. Composition of “operations” or “functions” is almost always associative, but not always commutative: for example, putting your shoes on before your socks doesn’t work the same as putting the socks on first.
• The operation has an identity element (which we usually write as $1$ or $1_G$ when we want to specify that we mean the identity for the group $G$, but sometimes other notations, like $0$ or $e$ are useful). The identity element “does nothing”p in the sense that $1 \cdot g = g \cdot 1 = g$ for every (“no matter what”) element $g$ in $G$.
• Every element $g$ has an inverse, i.e. an element $g^{-1}$ such that $g \cdot g^{-1} = g^{-1} \cdot g = 1$. Every symmetry of an object leaves it “the same” in some sense, so there must be a way to “undo” it.

The integers, rational, real and complex numbers, (all with the operation of addition) are groups. Invertible matrices, nonzero rational, real and complex numbers (all with the operation of multiplication) are groups. If you have a shape (a topological space, or some more geometric object like a polygon, a polytope or polyhedron, a Riemannian manifold, a cell complex, a graph, etc.), you might consider its group of homeomorphisms (or diffeomorphisms or isometries or automorphisms), homeomorphisms or homotopy equivalences up to homotopy (or isotopy), or other ways of associating a group to it.

If you have a group $G$ and a set $S$ (which might have more structure than just being a set), an action of $G$ on $S$ is a way of taking an element of $G$ and a point of $S$ and yielding another point of $S$ compatible with the group operation. Formally you can think of this as a function $G \times S \to S$, usually written $g.s$ (or $s.g$, more about this in a second). The action is a left action if $g.(h.s) = (gh).s$ and a right action if $g.(h.s) = (hg).s$, which looks weird at first until you write it as $(s.h).g = s.(hg)$.

Where do they show up?

In a young mathematician’s life in the US, groups usually show up for the first time in college as part of a core math major course with “Algebra” in its title; usually a student takes this class after they have finished their school’s calculus sequence, but before (or concurrently as) they take “elective” math major classes.

The algebra class might spread across two semesters, and usually covers “groups”, “rings” and “fields”. Groups and rings take their name in English from similar words in German. “Fields”, as far as I can tell, take their name from French—in German, the same concept uses the word “Körper”, which means “body” in english. Given the idea of a “crime ring” (cf. “The Bling Ring”), in college I liked to think of groups and rings as collections of little guys united with a common purpose.

Within academic mathematics, the group concept is truly foundational; every mathematical object exists in a context (we usually say a “category”) in which there is a natural sense of structure-preserving symmetry, which we usually abbreviate as “automorphism”, although as above, some notions of automorphism have special words. The collection of automorphisms of a mathematical object is always a group, and dealing with this collection of automorphisms is fundamental to understanding it. For example, computer scientists know that if you have two (finite) graphs, say with vertices and edges, the “graph isomorphism problem”—that is, if you have two graphs, how do you know that they are the same—is a difficult (in the sense of algorithmic difficulty; time-consuming) problem to solve. Part of the problem is that the “typical” graph may have many many automorphisms, and taking into account these automorphisms is a builtin part of the problem.

Do they have practical applications?

Sure! One easy answer here is “cryptography”. One source of mathematically difficult algorithmic processes is called the “word problem” (not to be confused with “word problems” in the sense of “Sally has three apples, …”): given two ordered lists of elements of a group, how can you tell that they multiply together to give the same element of that group? Or put another way, given $g_1,\ldots,g_n$, do you know whether $g_1\cdots g_n = 1$ or not? The “purely formal” product $g_1\cdots g_n$, when it is made up of elements chosen from a (typically finite) “alphabet” is called a “word”. It turns out that there is no general (this is a result named after Adian and Rabin) algorithm that will tell you the answer. What’s worse, Céjtin (as explained to me by this paper of Collins which has a lovely tone—kind of gleeful smashing of hopes) produced an example which you can write down with only five “generators” and seven “relations” which itself provably has algorithmically unsolvable word problem.

I recognize the use for cryptography in the world, but I have to confess that it does not spark joy for me personally. Accounting for symmetries, though, is an important part of the process of living life with objects real or natural or constructed or imagined. I don’t know if that counts as a practical application, but it does show that the concept is really fundamentally relevant to everyday life.

Where can I read more?

I have a copy of Paolo Aluffi’s Algebra: Chapter 0, and I would love to teach my university’s Algebra class from it. I didn’t learn from a textbook: my Algebra teacher came in every day and lectured with clarity such that the notes I took became the only learning material I needed. Another standard text is Dummit and Foote’s Abstract Algebra.

For geometric group theory, I love Groups, Graphs and Trees by John Meier, and Office Hours with a Geometric Group Theorist edited by Matt Clay and Dan Margalit.

I’d also like to start linking mathematics posts of mine together, but this post forms a link at the bottom of the chain.