TITLE: The number of commuting m-tuples is divisible by order of group: Improvements?
QUESTION [19 upvotes]: The number of commuting pairs of elements in finite group G is equal to the product $k(G)*|G|$ (see MO271757 ) where $k(G)$ is the number of conjugacy classes. Thus it is is divisible by $|G|$ (the number of elements of $G$). 
That divisibility also follows from a theorem by L. Solomon, stated below.
Question 0: It seems that the number of commuting $m$-tuples $c_m(G)$ is also divisible by $|G|$ for any $m$; is this correct? It appears to follow from a result cited in  Klyachko, Mkrtchyan (details below).
Question 1: Does the ratio $c_m(G)/|G|$ have some group theoretic interpretation for $m>2$?  (When $m=2$, this is the number of conjugacy classes).
If the group is abelian, then obviously $c_m(G) = |G|^m$, so it is divisible by a very high power of $|G|$. 
Question 2: Is there any improvement possible for this type of divisibility by $|G|$ for  nilpotent or $p$-groups ? 
Remark: Any improvement cannot contradict the analogues of the 5/8
bound for general $m$: $c_{m+1}(G) \leq \frac{3 \cdot 2^m - 1}{2^{2m+1}} |G|^{m+1}$ by  Lescot (see MO108392). 

Reminder 
Let me state theorems cited in Klyachko & Mkrtchyan 2012 (found in MO98639), and apply it to our situation.

Solomon theorem [1969]. In any group, the number of solutions to a system of coefficient-free equations is divisible
by the order of this group if the number of equations is less than the number of unknowns.
Application Consider the equation  $xy=yx$ in a group---one equation, two unknowns---thus the number of solutions should be divisible by the order of the group. Hence the number of commuting pairs is divisible by the order of group.

Note that we cannot apply that theorem for the number of commuting triples, $m$-tuples, since the number of equations exceeds the number of unknowns.
So one needs a refinement, and that seemed to be known:

Gordon,Rodriguez-Villegas theorem arXiv:1105.6066. In any group, the number of solutions of a system of coefficient free
equations is divisible by the order of this group if the rank of the matrix composed of exponent sums of $i$th
unknown in $j$th equation is less than the number of unknowns.

(It is presented like this by Klyachko & Mkrtchyan 2012, it is not immediately clear (to me) how to extract this formulation from the original paper). 

Application Consider equations defining commuting $m$-tuples: $x_ix_jx_i^{-1}x_j^{-1} = 1$. The sums the of exponents is ZERO! Thus the rank of the matrix is zero and hence the theorem ensures that the number of solutions is divisible by order of $G$.

I hope that this correct, and that an expert can confirm it.

Motivation
I hope (see MO271752) that there should be a nice generating function,
 $$ \sum_{n\geq 0} \frac{|\mathrm {
commuting~} m\mathrm{-tuples~  in~ GL}(n,F_q)|}{|GL(n,F_q)|}  x^n = ??? $$
This is similar to the known,
  $$ \sum_{n\geq 0} \frac{|\mathrm {
commuting~} \mathrm{pairs~  in~ GL}(n,F_q)|}{|GL(n,F_q)|} x^n = \prod_{j\geq 1}
    \frac{1-x^j}{1-qx^j} $$
which would imply  divisibility results at least for 
$GL(n,F_q)$ (and some other groups too), so it is 
nice to have  support for such a belief. 
About other groups, it would be  quite interesting for me to know especially
about the group $UT(n,q)$ (unitriangular matrices over a finite field)---
by what power of $q$ the numbers $c_m(G)$
are divisible.

REPLY [7 votes]: Another strategy is to use induction on $m,$ there being nothing to do if $m=1.$ Note that $(x_{1},x_{2},\ldots,x_{m})$ is a commuting $m$-tuple if and only if $(x_{2},\ldots,x_{m})$ is a commuting $(m-1)$-tuple of $C_{G}(x_{1}).$ By induction then, for every choice of $x_{1},$ the number of completions of $x_{1}$ to a commuting $m$-tuple with first component $x_{1}$ is divisible by $|C_{G}(x_{1})|.$ It follows easily that given a conjugacy class $C$ of $G,$ the number of commuting $m$-tuples whose first component is an element of $C$ is divisible by $|G|.$ Since the conjugacy classes partition $G,$ the number of commuting $m$-tuples is indeed divisible by $|G|.$
Notice also that if $p$ is a prime which does not divide $|G|$, then the number of mutually commuting $p$-tuples of elements of $G$ is congruent to $|G|$ (mod $p$), since the cyclic permutation of order $p$ acts on such commuting $p$-tuples with $|G|$ fixed points (the ordered $p$-tuples with all components equal).