The commuting probability of a finite group is the proportion of pairs which commute. In 1977 Keith Joseph made three conjectures about the set
namely the following.
Conjecture 1 (Joseph’s conjectures)
- All limit points of are rational.
- is well ordered by .
- is closed.
Earlier this month I uploaded a preprint to the arxiv which proves the first two of these conjectures, and yesterday I gave a talk at the algebra seminar in Oxford about the proof. While preparing the talk I noticed that some aspects of the proof are simpler from an ultrafinitary perspective, basically because ultrafilters can be used to streamline epsilon management, and I gave one indication of this perspective during the talk. In this post I wish to lay out the ultrafinitary approach in greater detail.
Throughout this post we fix a nonprincipal ultrafilter , and we let be the ultrapower , where two elements of are considered equivalent iff they are equal in -almost-every coordinate. The elements of are called “nonstandard reals” or “hyperreals”. There is a principle at work in nonstandard analysis, possibly called Los’s theorem, which asserts, without going into the finer details, that all “first-order” things that you do with reals carry over in a natural way to the field of hyperreals, and everything more-or-less works just how you’d like it to. For instance, if and are hyperreals then naturally means that the inequality holds in -almost-every coordinate, and similarly the field operations of extend naturally, and with these definitions becomes a totally ordered field. We will seldom spell out so explicitly how to naturally extend first-order properties in this way.
An ultrafinite group is an ultraproduct of finite groups. Its order is a nonstandard natural number, and its commuting probability is a nonstandard real in the interval . Joseph's first two conjectures can now be stated together in the following way.
Theorem 2 The commuting probability of every ultrafinite group has the form , where is a standard rational and is a nonnegative infinitesimal.
Somewhat similarly, Joseph’s third conjecture can be stated in an ultrafinitary way as follows: For every ultrafinite group there is a finite group such that . Here is the standard part operation, which maps a finite hyperreal to the nearest real. When phrased in this way it resembles a known result about compact groups. Every compact group has a unique normalised Haar measure, so we have a naturally defined notion of commuting probability . However every compact group with has a finite-index abelian subgroup, and with a little more work one can actually find a finite group with . This is a theorem of Hofmann and Russo. Nevertheless, I find Joseph’s third conjecture rather hard to believe.
For us the most important theorem about commuting probability is a theorem of Peter Neumann, which states that if then every finite group such that has a normal subgroup such that and are both bounded in terms of . To prove the above theorem we need the following “amplified” version:
Theorem 3 (Neumann’s theorem, amplified, ultrafinitary version) Every ultrafinite group has an internal normal subgroup such that is finite and such that almost every pair such that is contained in .
Here if we say that is internal if is itself an ultraproduct of subsets , and “almost every” needs little clarification because the set of pairs in question is an internal subset of . (Otherwise we would need to introduce Loeb measure.)
Proof: If the theorem holds with , so we may assume . By Neumann’s theorem has an internal normal subgroup of finite index such that is finite. Since is finite there are only finitely many normal subgroups containing and each of them is internal, so we may find normal subgroups containing such that is finite, and which are maximal subject to these conditions.
Suppose that a positive proportion of pairs outside of satisfied . Then we could find , say with , such that for a positive proportion of we have . After a little commutator algebra one can show then that for a positive proportion of we have , or in other words that the centraliser
of in has finite index. But this implies that the largest normal subgroup contained in , namely
where is the normal subgroup of generated by and , also has finite index. Since certainly a classical theorem of Baer implies that
is finite, and hence that is finite, but this contradicts the maximality of and .
Hence almost every pair such that is contained in , and thus also in , so the theorem holds for .
Now let be any ultrafinite group and let be as in the theorem. Then
where is nonnegative and infinitesimal. Thus it suffices to show that has the form
whenever is finite. Note in this case that Hall’s theorem implies that the second centre
has finite index. One can complete the proof using a little duality theory of abelian groups, but the ultrafinite perspective adds little here so I refer the reader to my paper.
The other thing I noticed while preparing my talk is that the best lower bound I knew for the order type of , , is easy to improve to , just by remembering that is a subsemigroup of . In fact the order type of a well ordered subsemigroup of is heavily restricted: it’s either , , or for some ordinal . This observation reduces the possibilities for the order type of to . I have no idea which it is!