There's a notion of commutativity measure $P(G)$ of a finite group $G$ which is probably folklore: count commuting pairs in $G \times G$ and divide by $|G \times G|$. There are some results:

  1. $P(G) > \frac{\mathrm{log}_2 \mathrm{log}_2 |G|}{|G|}$ (Erdős, Turán '68)

  2. If $P(G) > \frac{11}{32}$, then $P(G) = 1, \frac{3}{8}, \frac{25}{64}, \frac{2}{5}, \frac{11}{27}, \frac{7}{16}$ or $\frac{2^{2k}+1}{2^{2k+1}}$ and $G$ is metabelian (Rusin '79)

  3. $P(G) \leq |G : \mathrm{Fit}(G)|^{-1/2}$ (Guralnick, Robinson '06, uses CSFG)
  4. $P(G)$ is multiplicative wrt direct products, submultiplicative wrt extensions and nondecreasing when passing to subgroups
  5. $P(G)$ is isoclinism invariant
  6. There are several threshold values such that group is nilpotent, supersolvable or solvable modulo product with finite number of exceptions and isoclinism if $P$ is above them.
  • Fix a class $\mathcal K$ of finite groups. Is the closure of $\{P(G)\}, G \in \mathcal K$ known for some class? AFAIK, there's a conjecture that $P(\mathrm{Fin}) \cup 0$ is closed and I vaguely remember a paper where it's proved that values are well-ordered above certain value.

In principle, analogous questions make sense for amenable groups (well, not only amenable groups, but also finite monoids, finite rings which are special case of previous... you name it, but I think it's usually either too hard or uninteresting for some reasons. For example, in the case of finite semigroups $P$ obtains every rational value (V. Ponomarenko, '0s).

It's also pretty natural to consider $P_g(G): \frac{|com^{-1}(g)|}{|G|^2}$, where $com: G \times G \to G, (a, b) \mapsto [a, b]$, so $P(G) = P_1(G)$. It's pretty easy to obtain $P_g(G) = \frac {1}{|G|} \sum_{\chi \in Irr(G)} \frac{\chi(g)}{\chi(1)}$ and that $P_g(G)$ is isoclinism invariant too.

  • What is the closure of ${P_g(G)}, g \in G, G \in \mathcal K$?

There's an extension of $P(G)$ in a form of $\mathcal P(G) = \{P_i(G)\} \in c_{00}(\mathbb Q \cap [0, 1])$; $P_k$ counts fraction of commuting $k$-tuples in $G^k$ and they are isoclinism invariants. Maybe uniform approach can shed some light on these things? Besides obvious generalization, there's one a bit sublter: let's count $k$-tuples for which commutator $[g_1, \dots, g_k]$ vanishes. I don't see why this should be isoclinism invariant, and it probably isn't.

So, why the title? Those sets of values teasingly resemble spectra of some (possibly unbounded) operators on a topological vector space. There's a

  • Main question: are they really spectra of operators with a human face?
  • 1
    $\begingroup$ Perhaps arxiv.org/abs/1411.0848 is relevant to some of your questions, although I don't recall all the details in that paper $\endgroup$ – Yemon Choi May 28 '17 at 5:16

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