Groups and sets

Standard algebra texts define a group to be a set of objects together with a binary operation on those objects (obeying familiar conditions) — or indeed, they define a group to be an ordered pair of a set and a suitable binary function (where ordered pairs and functions are to be treated as sets too, in familiar ways).

But what work is the conventional set talk here actually doing? — is it really doing much more than the set talk back in the days of ‘new math’? Can these sets be treated as Quinean virtual classes? Can we in fact perfectly well think of a group as some objects (plural — or at least one) equipped with a group operation, and drop the set talk in favour of common-or-garden plural locutions?

There are a number of general philosophical issues here about the commitments of various grades of set talk, and about the commitments of plural talk too. But there’s an interesting technical question: how far can we get in group theory before we have to call on substantial ideas from set theory?

Let’s sharpen that question: what issues are there that can readily be understood by a relative beginner in group theory that depend for their answers on set-theoretic matters (e.g. whose answers can depending on the ambient set theoretical principles we countenance?). It is well known, for example, that Shelah showed in 1974 that the Whitehead problem is independent of ZFC. But the beginning student is surely not going to understand the significance of that (indeed picking up a couple of heavyweight standard texts, Dummit and Foote’s Abstract Algebra and Aluffi’s Algebra: Chapter 0, neither even mention the Whitehead problem). So put aside problems at that level. To repeat, then, what student-accessible problems have answers dependent on set theoretic ideas?

One suggestion might be

“Given objects X, is there always a group (X, e, *) with those objects, and some identity e and some group operation *?”

The answer is positive if and only if the Axiom of Choice is true. Though I suppose here we might now have a serious debate about whether the Axiom of Choice is at bottom an essentially set-theoretic principle at all (after all, some plural logics have versions of choice, and then there are type-theoretic versions).

Here’s a perhaps better example. The following

“Take a group G, its automorphism group Aut(G), the automorphism group of that, i.e. Aut(Aut(G)), the automorphism group of that, i.e. Aut(Aut(Aut(G))), etc. Does this automorphism tower terminate (count it as terminating when successive groups are isomorphic)?”

Joel Hamkins has shown that the answer is yes, but the very same group can lead to towers with wildly different heights in different set theoretic universes. (Why should this be, on reflection, no great surprise? — which is not to diminish Hamkins theorem one jot! Because, as he points out, there’s a sense in which going from a group to its automorphism group is ‘going up a level’; and we can play forcing tricks as we go up the levels.)

OK that’s a lovely example. But what others are there? I asked this on math.stackexchange here and the question got a surprising number of up-votes (suggesting I’m not alone in  finding the issue interesting).

And I didn’t get any more problems which are as immediately student-accessible as the automorphism towers problem, rather re-inforcing my guess that you can get a pretty substantial way into group theory without really tangling with set theory (especially if choice principles are assigned to the logic side of the ledger rather than the intrinsically set-theoretic side). But I did get two nice suggestions a notch or two up in sophistication, which you might be interested to see (if you know some algebra).

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