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.
Best Answer
This is in answer to your second question. There is a note by Gustafson:
where he proves the result Ben mentions, viz. if $G$ is a finite nonabelian group, then $d(G) \leq 5/8$. He goes on to prove that the same result for the case where $G$ is a compact, Hausdorff topological group (endowed with the Haar measure).
While $d(G)$ has received some attention over the years (I think it was first mentioned in a paper of Erdos in the late 60s and there have been sporadic papers since then) very little seems to have been said about $d(G)$ where $G$ is an infinite group until recently. The basic results (most of which are analogous to the finite case) are proved in
Ben has already mentioned the nice paper of Levai and Pyber where it is proved that if $G$ is a profinite group and $d(G) > 0$, then $G$ is abelian-by-finite. This result is extended to all compact groups in a recent preprint by Hofmann and Russo. There is much more besides in this preprint, I'm still digesting it myself!