# Conceptually, what do abelian groups represent? [closed]

Monoids represent maps from some mathematical object to itself. Groups represent the automorphisms of some mathematical object. What do abelian groups represent?

One unsatisfying answer would be that they just represent automorphisms of nice enough mathematical objects. However it seems rather uncommon for the automorphisms of some object to form an abelian group unless they have very few automorphisms. Furthermore, the theory of abelian groups is very different from the theory of groups in general, and abelian groups appear frequently in other applications where they are not the automorphisms of some object in an obvious way, like homology groups.

This suggests that maybe its just a coincidence that abelian groups are groups, and that they represent something else. What do they represent?

• Why should some object represent another? If $A$ represents $B$ then what does $B$ represent?
– lulu
Commented May 24, 2023 at 15:10
• (Surprisingly?) the "group prototype", namely the set of all the bijections on a given set, under map composition, is non-abelian. Commented May 24, 2023 at 16:00
• I think the closest you’ll get to an answer is by looking at the history. Wikipedia has an article “History of group theory”; scroll down to the section on number theory. The article by Israel Kleiner (in the Wikipedia references) goes into more detail. Commented May 24, 2023 at 17:13
• Indeed, groups arise as homology groups, or fundamental groups, or symmetry groups, or as group of units of a ring and so on. But this does not suggest that "maybe it's just a coincidence that abelian groups are groups". This is a misconception. Commented May 24, 2023 at 17:15

I don’t think this is a well formed question. Groups can “represent” all sorts of things. To say that groups in general are just the automorphisms of an object detracts from the abstraction of group theory. Groups can represent automorphisms, but they can represent all sorts of other things. It is often useful to simultaneously consider one group in multiple distinct ways to learn more about them. To reduce that to only one way, automorphisms, is unhelpful.

Now, abelian groups are the same. It’s unhelpful to consider them as one thing. They don’t represent anything in particular, but are instead a useful abstraction of many things at once.

Broadly speaking, this is sort of “the point” of abstract algebra (Topics in Algebra, Herstein). We abstract these algebraic objects away from their specific contexts, and learn much more about them and their connections with other things in the process.

• Could you give examples of what things do abelian groups represent? Commented May 24, 2023 at 14:38

Abelian groups are the same thing as $$\mathbf Z$$-modules; in a group, $$(gh)^2 = g^2h^2$$ if and only if the group is abelian. So ask yourself what a module over a commutative ring conceptually represents. Then ask yourself what a module over a PID or a Dedekind domain or a field (i.e., a vector space) conceptually represents.

• I have trouble thinking what rings conceptually represent, since their definition requires both abelian groups (the topic of this question) and tensor products/bilinear maps, which i also have trouble understanding. Let alone modules over a ring... Commented May 24, 2023 at 14:49
• Just as groups doesn't represent anything in particular neither do rings. This is just an illustration of why your way of thinking is problematic. Commented May 24, 2023 at 14:57
• @Henriksupportsthecommunity What should one think about when they think of rings then? How to get intuition for them or motivation to study them if they don't represent anything in particular? Commented May 24, 2023 at 15:50
• When I was at university and studied rings I mostly though about rings when I was supposed to think about rings. To me rings are an abstraction adding a multiplication on top of a group (that has an addition), and motivation mostly came from them being fun to study. Commented May 24, 2023 at 18:40