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) ${\,}$

All limit points of ${\mathcal{P}}$ are rational.

${\mathcal{P}}$ is well ordered by ${>}$ .

${\{0\}\cup\mathcal{P}}$ 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!