Joseph’s conjectures on commuting probability, an ultrafinitary perspective

The commuting probability {\Pr(G)} of a finite group {G} is the proportion of pairs {(x,y)\in G^2} which commute. In 1977 Keith Joseph made three conjectures about the set

\displaystyle \mathcal{P} = \{\Pr(G) : G \text{ a finite group}\},

namely the following.

Conjecture 1 (Joseph’s conjectures) {\,}

  1. All limit points of {\mathcal{P}} are rational.
  2. {\mathcal{P}} is well ordered by {>}.
  3. {\{0\}\cup\mathcal{P}} 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 {u\in\beta\mathbf{N}\setminus\mathbf{N}}, and we let {\mathbf{R}^*} be the ultrapower {\mathbf{R}^\mathbf{N}/u}, where two elements of {\mathbf{R}^\mathbf{N}} are considered equivalent iff they are equal in {u}-almost-every coordinate. The elements of {\mathbf{R}^*} 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 {r} and {s} are hyperreals then {r<s} naturally means that the inequality holds in {u}-almost-every coordinate, and similarly the field operations of {\mathbf{R}} extend naturally, and with these definitions {\mathbf{R}^*} 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 {G} is an ultraproduct {\prod_u G_i = \prod_{i=1}^\infty G_i/u} of finite groups. Its order {|G| = (|G_i|)/u} is a nonstandard natural number, and its commuting probability {\Pr(G) = (\Pr(G_i))/u} is a nonstandard real in the interval {[0,1]}. Joseph's first two conjectures can now be stated together in the following way.

Theorem 2 The commuting probability of every ultrafinite group {G} has the form {q+\epsilon}, where {q} is a standard rational and {\epsilon} is a nonnegative infinitesimal.

Somewhat similarly, Joseph’s third conjecture can be stated in an ultrafinitary way as follows: For every ultrafinite group {G} there is a finite group {H} such that {\text{st}\Pr(G)=\Pr(H)}. Here {\text{st}} 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 {G} has a unique normalised Haar measure, so we have a naturally defined notion of commuting probability {\Pr(G)}. However every compact group {G} with {\Pr(G)>0} has a finite-index abelian subgroup, and with a little more work one can actually find a finite group {H} with {\Pr(G)=\Pr(H)}. 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 {\epsilon>0} then every finite group {G} such that {\Pr(G)\geq\epsilon} has a normal subgroup {H} such that {|G/H|} and {|[H,H]|} are both bounded in terms of {\epsilon}. To prove the above theorem we need the following “amplified” version:

Theorem 3 (Neumann’s theorem, amplified, ultrafinitary version) Every ultrafinite group {G} has an internal normal subgroup {H} such that {[H,H]} is finite and such that almost every pair {(x,y)\in G^2} such that {[x,y]\in[H,H]} is contained in {H^2}.

Here if {G = \prod_u G_i} we say that {S\subset G} is internal if {S} is itself an ultraproduct {\prod_u S_i} of subsets {S_i\subset G_i}, and “almost every” needs little clarification because the set of pairs in question is an internal subset of {G^2}. (Otherwise we would need to introduce Loeb measure.)

Proof: If {\text{st}\Pr(G)=0} the theorem holds with {H=1}, so we may assume {\text{st}\Pr(G)>0}. By Neumann’s theorem {G} has an internal normal subgroup {K_0} of finite index such that {[K_0,K_0]} is finite. Since {G/K_0} is finite there are only finitely many normal subgroups {K\leq G} containing {K_0} and each of them is internal, so we may find normal subgroups {K,L\leq G} containing {K_0} such that {[K,L]} is finite, and which are maximal subject to these conditions.

Suppose that a positive proportion of pairs {(x,y)\in G^2} outside of {K\times L} satisfied {[x,y]\in[K,L]}. Then we could find {(x,y)\in G^2\setminus (K\times L)}, say with {x\notin K}, such that for a positive proportion of {(k,l)\in K\times L} we have {[xk,yl]\in[K,L]}. After a little commutator algebra one can show then that for a positive proportion of {l\in L} we have {[x,l]\in[K,L]}, or in other words that the centraliser

\displaystyle N_0 = C_{L/[K,L]}(x) = C_{L/[K,L]}(\langle K,x\rangle)

of {x} in {L/[K,L]} has finite index. But this implies that the largest normal subgroup contained in {N_0}, namely

\displaystyle N = C_{L/[K,L]}(K'),

where {K'} is the normal subgroup of {G} generated by {K} and {x}, also has finite index. Since certainly {K\leq C_{K'/[K,L]}(L)} a classical theorem of Baer implies that

\displaystyle [K'/[K,L], L/[K,L]] = [K',L]/[K,L]

is finite, and hence that {[K',L]} is finite, but this contradicts the maximality of {K} and {L}.

Hence almost every pair {(x,y)\in G^2} such that {[x,y]\in[K,L]} is contained in {K\times L}, and thus also in {L\times K}, so the theorem holds for {H=K\cap L}. \Box

Now let {G} be any ultrafinite group {G} and let {H} be as in the theorem. Then

\displaystyle \Pr(G) = \frac1{|G/H|^2} \Pr(H) + \epsilon,

where {\epsilon} is nonnegative and infinitesimal. Thus it suffices to show that {\Pr(H)} has the form

\displaystyle (\text{standard rational}) + (\text{nonnegative infinitesimal})

whenever {[H,H]} is finite. Note in this case that Hall’s theorem implies that the second centre

\displaystyle Z_2(H) = \{h\in H : [h,H]\subset Z(H)\}

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 {\mathcal{P}}, {\omega^2}, is easy to improve to {\omega^\omega}, just by remembering that {\mathcal{P}} is a subsemigroup of {(0,1]}. In fact the order type of a well ordered subsemigroup of {(0,1]} is heavily restricted: it’s either {0}, {1}, or {\omega^{\omega^\alpha}} for some ordinal {\alpha}. This observation reduces the possibilities for the order type of {\mathcal{P}} to {\{\omega^\omega,\omega^{\omega^2}\}}. I have no idea which it is!

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