TITLE: Degree of commutativity of finite groups and subgroups
QUESTION [6 upvotes]: Recently I started reading some articles about the
degree of commutativity of finite groups. I have some questions:

In "Subgroup commutativity degrees of finite groups" Tarnauceanu
proposes the following formula for calculating the degree of
commutativity of subgroups of a finite group G:

$$ \mathrm{sd}(G) = \frac{\left| \left\{(H,K) \in \mathscr{L}(G)^2:HK=KH\right\}\right|}{\left|\mathscr{L}(G)\right|^2}, $$
He proves that if $G_1, G_2, \ldots , G_n$ are finite groups of coprime order, then
$$ \mathrm{sd}\left(\times_{i=1}^{n}G_{i}\right)=\prod_{i=1}^{n}\mathrm{sd}(G_i). $$
My first question is about what happens if we omit the hypothesis that $G_i$ have coprime orders, that is, if there exists some estimate for
$$\mathrm{sd}\left(\times_{i=1}^{n}G_{i}\right)$$ 
in terms of $\mathrm{sd}(G_i)$.

In "Central Extensions and Commutativity Degree" Lescot proposes
the following formula for calculating the degree of commutativity of a
finite group $G$:

$$\mathrm{d}(G)=\frac{1}{|G|^2} \left|\left\{(x,y) \in G^2:xy=yx\right\}\right|.$$
My second question is about the order of the group $G$. Is there
any theory for the case when $G$ is infinite? For example, $G$ might be a
group equipped with a Haar measure. I have found no literature about this
case.
Remark: Sorry for the obscurity of my question, the first question is: For $G_1,\ldots,G_n$ arbitrary groups, is there any estimate for the degree of commutativity of subgroups of $G=direct~product~of~the~groups~ G_i$ in terms of the degree of commutativity of the groups $G_i$? My second question was partially answered. A friend showed me the following article: tmu.ac.ir/salg20/talks/Rezaei.pdf.

REPLY [3 votes]: I am quite late on this one, but here's a partial answer to your first question.
First, a couple of comments: 

I take the view that the term "commutativity" is unfortunate when applied to subgroups; permutability is more appropriate.
There is really no need to talk about the subgroup lattice of the group $G$. Lattices have structure, but here we only need their size, so set of subgroups,
which you may denote by $\mathrm{s}(G)$ for instance, would be just as good.

The coprime orders case you mention (Proposition 2.2 of the original article) is, of course, valid. If $\gcd(|A|,|B|)=1$, then every subgroup of $A \times B$ is
a subproduct, i.e., a direct product of subgroups of the two factors. This follows directly from Goursat's lemma; you can have a look at 

D.D. Anderson and V. Camillo, Subgroups of direct products of groups, ideals and subrings of direct products of rings, and Goursat’s lemma, Rings, modules and representations. International conference on rings and things in honor of Carl Faith and Barbara Osofsky, Zanesville, OH, USA, June 15–17, 2007, American Mathematical Society, 2009, pp. 1–12.

for an excellent account of this useful result (and some generalisations). As far as I can tell, Tarnauceanu does not offer any justification for Proposition 2.2 in his paper, that is why I mention this.

The following properties are all satisfied by $\mathrm{d}$, but not by $\mathrm{sd}$:

If $G_1$, $G_2$ are groups then $\mathrm{d}(G_1 \times G_2) = \mathrm{d}(G_1) \cdot \mathrm{d}(G_2)$.
If $N$ is a normal subgroup of $G$ then $\mathrm{d}(G) \leq \mathrm{d}(N) \cdot \mathrm{d}(G/N)$.
If $H$ is a subgroup of $G$ then $\mathrm{d}(H) \geq \mathrm{d}(G)$.


For instance, the dicyclic group of order 12 with presentation 
$$\mathrm{Dic}_3 = \left\langle a, x : a^{6} = 1, x^2 = a^3, x^{-1}ax = a^{-1}\right\rangle,$$
has $\mathrm{sd}\left( \mathrm{Dic}_3\right) = 29/32$ and a normal subgroup of order 2 with quotient isomorphic to $\Sigma_3$, 
but $\mathrm{sd}\left( \Sigma_3\right) = 5/6$, thus 2. is not true. Then the group of order 16 which is a central product of $D_8$ and $C_4$ over a common cyclic central subgroup of order 2 has subgroup permutability degree equal to $505/529$, but $\mathrm{sd}\left( D_8 \right) = 23/25$, thereby disproving 3. 
Finally, $\mathrm{sd}\left( C_2 \times \Sigma_3\right) = \mathrm{sd}\left(D_{12} \right) = 101/128$, but $\mathrm{sd}\left( \Sigma_3\right) = 5/6$, 
hence 1. does not hold either.