Active and passive groups
Posted by Martin Orr on Thursday, 02 October 2008 at 19:13
I have been studying category theory recently, and revising my idea of what a group is. There are two ways of thinking about groups, which I shall call active and passive. I have tended to almost exclusively think about groups passively, but I have realised that treating groups as active is useful too.
First, the passive definition of a group is what you would probably write down if someone asked you to define a group: a set with a binary operation satisfying axioms of associativity, identity and inverses. Here the elements of the group are thought of as static objects.
The active approach is to think about a group acting on some object. Of course, there is an abstract theory of (passive) group actions, which is sometimes useful (e.g. in representation theory, where you want to consider many different actions of the same group) but for many groups, it goes deeper than this: the group is defined by its action on some space e.g. the group of linear transformations of a vector space, the Galois group of a field extension. A formal definition of an active group on a set X would be:
A set G of bijections
which is closed under composition and taking inverses, and which contains the identity.
Thinking about an active group as a passive group is easy: just take the functions in G as elements and let the operation on them be "composition". This is automatically associative.
Less obviously, you can think of any passive group G as an active group. According to Tom Körner's notes, "The proof of the next result is simple but faintly Zen." I don't think this is really true: you have to choose some set for G to act on. Fixed things like a one-point set don't work, so the only set you have available is the elements of G itself. The simplest function 
This is usually taught in a first course on groups under the name of "Cayley's theorem", and I remember thinking it was not very interesting. If you view it as telling you that you can view all groups from this second, active perspective, rather than as just a formal isomorphism, then it is more important.
The group of units mod n is a number theoretic group which it is probably rarely useful to think about actively, but consider the following elementary proof of the Fermat's Little Theorem:
Let a be coprime to p, a prime. Write down the integers 1, 2, ..., p-1. Underneath them write down a, 2a, ..., (p-1)a (mod p). Because a is invertible mod p, the second row is a permutation of the first. So multiplying the top row together is the same as multiplying together the bottom row, and regrouping a little we have
. (p-1)! is invertible so we get.
This proof begins by considering the operation "multiply by a" on the group itself, which is an example of thinking actively. And incidentally it extends straightforwardly to a proof (not the standard one) that 
Introductions to group theory usually introduce both perspectives at the beginning, but I think that I (and quite possibly many people) tend to focus on passive groups and ignore the active perspective. I don't claim that it is always right to think of groups actively, but it is useful to be aware that the possibility is there, even for groups that are defined passively.
There's another notion, which I consider valuable.
Define a "group with action" to be a pair consisting of a group G, a set X and an action of G on X.
These form the objects of a category; morphisms from (G,X) to (H,Y) consist of a group homomorphism G -> H and a set map X -> Y which are compatible in the sense that the obvious square of morphisms (from GxX to Y) commutes.
Then "active groups" are (essentially) the subcategory where the action is an injection.
I guess the point is that restricting to injections is category-theoretically a bit weird, and sometimes it's better to work in this slightly more general setting.
This idea can be imitated all over the place; there's a category of "groups with linear representations", a category of "rings with modules" and so on.
I hadn't really considered making "active groups" into a category, but I agree that if you want to do that then "groups with actions" are more natural.
And your explicitly defining a group with action as a pair (G,X) brings out another thing I passed over in my explanation: that there are many active groups/groups with actions corresponding to a single passive group (up to isomorphism).
The thing was that I was using active groups as a way to define a group by its action, instead of starting with the group and then considering its actions. So for that, we do need the action to be injective.