Recently I started reading some articles about the
degree of commutativity of finite groups. I have some questions:

1. 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 $\mathrm{sd}(G_i)$.

2. 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: My second question was partially answered. A friend showed me the following article: tmu.ac.ir/salg20/talks/Rezaei.pdf.