What the hell is a group?Abstract. Group theory is often described as the “study of symmetry”. But how does that relate to the definition of groups? A symmetry is essentially a bijection; through the lens of Cayley’s theorem and generating sets, a group is a collection of symmetries (bijections) taken together with their compositions and inverses.
Symmetries and BijectionsTake a look at this triangle:
To be clear, when I say “this triangle”, I thinking of the triangle as a collection of points . As such, we can transform it the same way we can transform anything in ; namely, with functions . We can, for instance,
Dilate the triangle about its center by a factor of :
Reflect the triangle vertically about its center ...
... or along another line
Rotate by ...
... or by
Notice that two of these transformations don’t seem to “do anything”. Namely, the second reflection and rotation leave looking exactly the same as it started. These fixpoints are witnesses to the symmetries of regular triangles! The reason that the second reflection has no effect on is because the reflection is about an axis of symmetry; likewise, the reason that a rotation by has no effect on is because of the radial symmetry of regular triangles.
Generally, if we can find a transformation such that all it does is shuffle the points of around (ie, where ), then we figure that this is witness to some symmetry of our shape ; as such, we call a symmetry1. Note that this condition is exactly the condition that is a bijective function from to ! In other words, symmetries are exactly bijections.
Since we are considering to be a locus of points, this technically means that there are many symmetries on a triangle which “feel wrong”. For instance, we could swap two points of the triangle—that’s a symmetry! What’s going on here? Symmetries care only about the set structure of what we are working with. The fact that the points in form three lines is irrelevant to what is and isn’t a symmetry; to be a symmetry one need only preserve the collection of points. But when we think of triangles we don’t want consider all this nonsense to be a symmetry. We can solve this in one of two ways. The first is to restrict ourselves to talking about isometries (that is, distance-preserving symmetries). The second way is to not consider a triangle to be a locus of points, but to be defined only by its vertices, with “understood” edges between them. For the rest of this page, though, we’ll just sweep this issue under the rug.We can think of group theory, loosely speaking, as the study of objects (for instance, ) taken alongside a set of symmetries on those objects.
CompletionSo group theory is about symmetries and by “symmetry” we mean “bijection”. Ok, but what can we do with that? Say I hand you the object alongside the following two symmetries over :
What interesting things can you tell me about and and ? Well, here’s something interesting. The vertical reflection:
Can be constructed by composing and :
Likewise, we can construct:
The counter-clockwise rotation as
The “do nothing” symmetry as
As it turns out, every symmetry2 of the square can be expressed as some combination of and !
The reflection over the falling diagonal as
By which I mean in particular isometries (length-preserving bijections)I give this example for a reason. We reached an interesting conclusion, and in order to do so, it was not enough to talk about and on their own. Instead, we worked with compositions of and . Likewise, group theory is not interested in viewing symmetries in isolation. Rather, given a collection of symmetries over some set , a group theorist will be interested not only in symmetries but also:
Compositions of symmetries
The identity bijection
Altogether, let us denote such a “completed” set of symmetries3 by . Again, we are interested in rather than just simply because we can do interesting things with it!
Inverses of symmetries (and compositions thereof)
What I’m giving here is essentially a contextualized presentation of the concept of group generating setsActually, what a group is is a completed collection of symmetries over some object . Kind of....
Cayley’s TheoremThinking of groups as completed collections of symmetries is loosely reasonable, but pedantically wrong. Let’s bridge that gap. Notice that when we performed the “completion” from to , we did so with respect to composition . That is, we asked for:
Compositions of elements of ; that is, applications of
An identity element with respect to ; that is, the symmetry where for every we have .
As it turns out, these three conditions—now that we have relativized them with respect to the operator —produce a set of axioms which essentially form their own characterization of a group! Precisely, Take a 2-tuple consisting of a set and any binary operator on , and assume that:
Inverses of elements in with respect to ; that is, for each the value exhibiting
contains an identity; that is,
Then there exists a completed collection of symmetries over such that behaves on in the same way that behaves on 4. This is known as Cayley’s Theorem.
contains inverses wrt ; that is,
By “behaves in the same way” I mean in particular that there exists a bijection such that:Since each such 2-tuple acts like a completed collection of symmetries, then the above three axioms act to give an abstract characterization of what a group is. In fact, these three axioms are the modern definition of “group”!
In other words, there is a way to go between and which “respects” the relevant structure. (If you’re not familiar with category theory, and this doesn’t make much sense, that’s fine)
RecapA “symmetry” on an object is just a bijection . Group theory is essentially the study of such symmetries. A single symmetry is typically not interesting in isolation, so we study collections of symmetries taken together with their inverses and compositions thereof. Modern treatments of group theory do not speak about symmetries directly, but instead study groups via Cayley’s theorem through a more abstract notion defined by the modern group axioms.
ExamplesThe only examples of groups I have given so far are the symmetries on the triangle and the square 5. These are fine examples, but give a very narrow view of group theory. There are many groups not directly related to polygons!
These are called dihedral groups, if you’re curious!To rectify the situation, here are some examples of groups with different “feels” to them.
. This is the group of all bijections on a set . This should feel familiar!
. This is the group of all translations on ; that is, functions of the form for some . We can consider a translation by to be a witness to the fact that sliding by produces .
. This is the group of dilations on ; that is, functions of the form for some . We can consider a dilation by to be a witness to the fact that dilating by produces .
. Likewise, this is the group of all nontrivial dilation-rotation compositions on
. This is the group of translations on , but where “wraps around” to zero at ; think of a clock
. This is the group of invertible square matrices of size .
Soft language description of groups:
A group is a collection of things, each of which represents an action which has a disjoint reversala collection of things — a set each of which represents an action — cayleys theorem which has a ... reversal — group elements have inverses disjoint revesal — group inverses must be two-sided.
Sufficient for this is that an element commutes with its inverse, which IMO is more informative: we can think of two actions commuting as them being ‘disjoint’, so here we’re asking for all actions to have ‘disjoint’ inverses For an example of non-‘disjoint’ inverses, consider the following. Starting with a finite alphabet and forming the set of words over this alphabet. Then every append action would have the inverse here is the post-inverse of , but it is somewhat unsatisfying because it in some sense “only makes sense” as an action if it is performed immediately after . this is in some sense “bad”, and the antidote is requiring disjointness (commutativity)
Little dump of how I might explain a group at this moment to someone (2023-01-31):
A group is a collection of actions, each of which may uniformly be reversed. Though this has four technical terms in it, for the most part you already know what I mean. By collection I mean, for the most part, a collection; by action I mean, for the most part, an action, and by reverse I mean, for the most part, reverse. But let’s dig in.actions — By actions I mean “something that can be done” or, more precisely, “something that can have an effect on the universe”. Literally: you opening a door. That’s an action. Doing groceries. That’s an action. etc. etc. Well, kind of. We are going to place some footnotes on these actions. The first is this:Action footnote number one: an action may live in a reduced universe. The exact impact of any real-world action depends on a massive, massive number of factors. Opening a door, for instance. The exact force with which the door is opened, and the angle at which it’s opened, the air humidity, the grain of wood: all of these play into exactly how the action of opening the door affects the door, what angle it ends up at, etc, etc. Trying to talk about all those factors at once is hard. Mathematics elects to simply not, preferring to view only a single “slice” of the universe at a time. We may, for instance, ignore all the little details about the door’s position and pay attention to only this: is it open or closed? And we may further demand that the answer always be one of “open” or “closed” (which gets hairy) Or say we are performing a statistical analysis of some medical trials. Each individual person in the trial contains a multitude of complexity, but we forget most of it: perhaps we keep height, weight, age, ethnicity, etc.Action footnote number two: an action must be deterministic. Sorry, lovers of free will. Also, the action must be fully determined by the slice of the universe we’ve chosen. We cannot choose an action which “opens the door if it’s sunny” unless our universe includes both the door and the weather.Action footnote number three: an action must be unconditional. It’s common in language to talk about actions which make assumptions on the state of the universe. One might say, for instance, “please open the cupboard and hand me the salt”. Completing this action rests on the assumption that the cupboard exists and contains salt. This will not do for us. We would either need to place that action inside of a universe where the cupboard and salt definitely exist, or have our action account for the case where either conditions fail. An action must, ahead-of-time, account for all possibilities.reversed — this is what you think it is. An action can be reversed if there’s another action that can be taken afterwards which undoes everything. For instance. say our slice of the universe contains only a door and concerns itself only with whether it’s open or closed. Let denote the action of opening the door. Then can be reversed as follows. If the door is open to begin, then we ‘reverse’ by doing nothing (there’s nothing to undo). If the door is closed to begin, then we reverse by closing the door.Hence a group is a collection (set) of actions (elements; operation is associative) which can be reversed (have right-inverses) uniformly (which commute) — a group obviously in reality we’d want to expound upon these definitions a whole bunch.uniformly reversed — this one is going to be more difficult. Say we are in the prep phase of a recipe, and we have bowls on the counter with various ingredients. Our slice of the universe considers only what bowls are on the counter, and what ingredients are in these bowls. Amount of ingredients and number of dishes in the sink are ignored. Now let denote the action of adding a bowl of milk to the counter. We can reverse this action by removing a bowl of milk from the counter. Call this . Then performing right after completely undoes the , just as we want. But recall that actions are required to be unconditional: the action must also have instructions for what to do if there is no milk to remove. We can resolve this simply by asking that do nothing in the cases where there is no milk to remove. Perhaps this feels a little unsatisfactory, though: amending this way feels like something we shouldn’t even have to think about. To that end, we want reversals to be uniform, which is to say, they don’t have to worry about these edge cases. In particular, we ask that an action’s reversal can be done either before or after the action and either way it will negate the action. In the case of the recipe, we can do this by expanding our universe to also allow us to place sticky notes on the counter which say “if you are going to add a milk, please don’t”. Then will remove a milk if one is present, or add a sticky note otherwise. Mathematically we say that an action and its reversal must commute. (alt. justification — we want actions and their removals to be ‘disjoint’ in that they don’t interact with each other. that is, they commute)