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## Groups vs. abelian groups

A few weeks ago on MathOverflow Greg Muller asked, “why do groups and abelian groups feel so different?” The answers were very interesting and came from several different perspectives, but I still don’t feel as if the question was resolved. So I’ll try to synthesize and summarize some of the answers and hopefully something will be clearer in the end.

Morphisms

Groups naturally arise as automorphism groups of objects in categories. From this perspective abelian groups should just correspond to objects with particularly simple symmetries. But abelian groups have their own special type of symmetry: the pointwise sum of two homomorphisms is also a homomorphism, so $\text{Ab}$ is enriched over itself, something that doesn’t hold at all for $\text{Grp}$. What this implies is that the automorphisms of an abelian group have two composition structures: one coming from pointwise addition and another coming from composition. This is the “best” definition of a ring, at least in my opinion.

This observation also immediately tells us why $\mathbb{Z}$ is a special ring: it is the initial object in $\text{Ring}$, since every abelian group has a $\mathbb{Z}$-action given by $x \mapsto x^n$. From here one is readily led to the “best” definition of a module, exactly analogous to how one might go from sets to permutations to groups to group actions. Now we can think of abelian groups as $\mathbb{Z}$-modules and the action $x \mapsto x^n$ as a distinguished set of automorphisms of any abelian group. For some abelian groups these are the only automorphisms.

Non-abelian groups, on the other hand, have a totally different set of distinguished automorphisms: the inner automorphisms $x \mapsto gxg^{-1}$. Indeed, one way to define a non-abelian group is as a group with nontrivial inner automorphisms, and for some non-abelian groups these are the only automorphisms. In other words, while the structure of abelian groups is controlled in a strong way by the structure of $\mathbb{Z}$ (this is basically the content of the structure theorem), there is no such controlling object for general groups.

Perhaps precisely because their structure is so well-controlled, abelian groups lead to a lot of useful constructions in mathematics, such as abelian categories (the basic tool of homological algebra). Even to study groups one often passes to the category of modules over the group algebra!

What’s not clear to me is the following: should I think of abelianness as a powerful tool or as the easier version of a more difficult but more powerful theory of “non-abelian categories”?

A 2-categorical perspective

One particular complaint Greg had was that abelian groups rarely arise, in practice, as automorphism groups of objects. There is a way, however, one can get abelian groups “naturally” acting on things. The Eckmann-Hilton argument shows that two monoid structures on a set which are homomorphisms of each other must in fact be the same commutative monoid structure, and has the following consequences:

• The higher homotopy groups are abelian.
• The $2$-morphisms in a $2$-category with one object and one morphism form a commutative monoid. If the $2$-morphisms are invertible, they form an abelian group.

The latter is due to the fact that $2$-categories have two types of composition. This seems to have important philosophical implications; for example, awhile back at the n-Category cafe Bruce Bartlett asked a similar question about the difference between commutative and noncommutative algebras from an $n$-categorical point of view. It seems as if to understand commutativity one should really move up the $n$-categorical ladder; as Scott Carnahan puts it, if groups act on objects, then 2-groups act on categories, and the automorphisms of the identity functor automatically form an abelian group.

The big question, of course, is whether the abelian groups that naturally arise in mathematics can actually be fruitfully thought of in this way. Anyone have any thoughts?

### 8 Responses

1. I think part of what you’re getting at in the 2-categories section is that AbGroups is an abelian category but Groups is not(not even additive) and since if we consider the 2-category of an abelian category we get an abelian category(morphisms are Z-matrices since Hom sets are abelian groups). Thus moving up this ladder, the AbGroups look similar(the same?) but Groups does not. This may be similar to what your saying but at least this is something that _I_ find interesting about them.

B.t.w. nice blog, I enjoy it.

2. “should I think of abelianness as a powerful tool or as the easier version of a more difficult but more powerful theory of “non-abelian categories”?”

Well, obviously, the answer depends on context. For example, if you’re studying the automorphism group of some class of mathematical objects, and manage to prove that it’s always abelian, then that (may) turn into a powerful tool for telling you something about your objects.

You can ask similar questions in many areas of mathematics: e.g. replace abelianness w/ “field/vector space” and non-abelianness with “ring/module”. (Keeping in mind that modules over a group ring are morally equivalent to group representations) I don’t think of linear algebra as just an easier version of the representation theory of groups. I think of linear algebra as somehow more fundamental…but maybe I just haven’t thought about it too much.

• I think you’ve slightly misunderstood my question. What I mean is this: homological algebra is all about abelian categories, so when I say “abelianness” I’m including modules over rings. Is there a “non-abelian homological algebra” on “non-abelian categories” which generalizes homological algebra?

3. Well, here’s something I was thinking, although maybe it’s equivalent to what you said.

Groups are categories, right? Specifically, they’re groupoids that are monoids. So it makes sense to define functors between groups, which happen to coincide with homomorphisms. Now two functors (in the categorical setting)/homomorphisms (in the group-theoretic setting), which we’ll call $f, g: G \rightarrow H$, are naturally isomorphic iff there’s some element $a \in H$ with $f = aga^{-1}$. Now if H is abelian, then this happens iff $f = g$, so two parallel functors are naturally isomorphic iff they’re not just equivalent, but actually “the same.” But if H is nonabelian, there are plenty of natural transformations between different morphisms.

There’s gotta be a way of phrasing this categorically — I think we’re saying that the 2-category of abelian groups is a strict 2-groupoid, whereas the 2-category of groups is only a weak 2-groupoid? Again, this “feels” like it should tell us why the higher homotopy groups are abelian, although I don’t really see how.

• I interpret it as meaning that the Cartesian closure of $\text{Grp}$ is $\text{Gpd}$, but I don’t know if this is true or meaningful.

• Is $Gpd$ really cartesian closed? I think I see what you’re going for, though, although at best this seems like a 1-categorical shadow of a 2-categorical statement.