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.
