Hi everybody! 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:

$$sd(G)= \displaystyle \frac{1}{|\mathscr{L}(G)|^2}\ |\{(H,K)\in\mathscr{L}(G)^2\ |\ HK=KH\}|$$

, he proves in this work that if

$$G_1, G_2, \ldots , G_n$$ are finite groups of coprime order than

$$ sd(\times_{i=1}^{n}G_{i})=\prod_{i=1}^{n}sd(G_i) $$

My first question is about what happens if we omit the hypothesis of $G_i$ have order coprime, that is, exists some estimative for $$sd(\times_{i=1}^{n}G_{i})$$ is there any estimate in terms of $sd(G_i)$

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

$$d(G)=\frac{1}{|G|^2}|\{(x,y)\in G\times G\;|\;xy=yx\}|.$$

My second questions is about of order of group $G$, that is, there is any theory for the case that G is infinite? For example if G is a group equipped with a Haar measure? I found no literature about this case that G is infinite, there is some technical difficulty in trying to do something analogous in 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 articles: tmu.ac.ir/salg20/talks/Rezaei.pdf.