How nearly abelian are nilpotent groups?

Question

It is not uncommon to read that "nilpotent groups are 'close to abelian'."^1,2 Can this sentiment be made precise in the sense of the Turán and Erdős definition of "the probability that two elements of $G$ commute," discussed in the MO question, "Measures of non-abelian-ness"?

More precisely,

Q. What the probability that two randomly selected elements of a nilpotent group commute? What is the max and min over all nilpotent groups?

For example, the nilpotent Quaternion group $Q_8$ is $62.5\%$ abelian, if I calculated correctly:

---

         
          ^{$Q_8$: $24$ out of $64$ entries are non-abelian. So it is $40/64=62.5\%$ abelian.}

---

Update. @BenjaminSteinberg immediately observed that the $Q_8$ example establishes the upper bound of $5/8$, and shortly thereafter cited literature that shows that there is no positive lowerbound.

---

¹Princeton Companion to Mathematics, "Gromov's Polynomial-Growth Theorem," p.702.: 'nilpotent groups are "close to abelian."'

²Wikipedia: Nilpotent group: 'a nilpotent group is a group that is "almost abelian."'

Answer 1

Here are a few general remarks (concerning finite groups). The relevant results can mostly be found in the 2006 Journal of Algebra paper by Bob Guralnick and myself (a link to a preprint version is given in one of Benjamin Steinberg's comments). An extra-special $p$-group of order $p^{3}$ has commuting probability $\frac{p^{2}+p-1}{p^{3}},$ which is only slightly more than $\frac{1}{p},$ so according to that measure, for a large prime $p,$ such a group is quite far from Abelian. The asymptotic behaviour of the number of $p$-groups of order $p^{n}$ was shown by C. Sims (and maybe G. Higman) to be much the same as that of the number of $p$-groups of nilpotence class $2$ and order $p^{n},$ so perhaps it is not surprising that $p$-groups of class $2$ already exhibit a high measure of non-commutativity (this is also borne out by a theorem of P. Neumann, of which Guralnick and I were unaware when we wrote our paper, which was pointed out by a comment by Sean Eberhard to an earlier MO question on this topic-in fact, the question mentioned in Joseph O'Rourke's first comment). Note that the largest Abelian subgroup of an extraspecial $p$-group of order $p^{2n+1}$ has order $p^{n+1},$ which is not much more than the square root of the group order.

But perhaps such examples illustrate that the commuting probability is too crude a measure of non-Abelianness, since a group of nilpotence class $2$ is in some ways not too far from Abelian.