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Hi everybody! 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:

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

, 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 $sd(G_i)$

  1. 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.

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What does that fancy L mean, in your first display? –  Gerry Myerson Mar 10 '11 at 23:35
    
$\mathscr{L}(G)$ is the subgroup lattice of G –  Juan Valdez Mar 11 '11 at 0:25
    
In the title, "Commutativity degree" will be better than "Degree of Commutativity"; "Degree of commutativity" is an important (and totally different) concept in p-groups of maximal class, and it is an interesting area of research in theory of finite p-groups. –  RDK Jan 17 '13 at 4:47
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3 Answers

up vote 4 down vote accepted

This is in answer to your second question. There is a note by Gustafson:

  • MR0327901 (48 #6243) Gustafson, W. H. What is the probability that two group elements commute? Amer. Math. Monthly 80 (1973), 1031–1034.

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

  • MR2558527 (2010m:22003) Rezaei, Rashid; Erfanian, Ahmad(IR-MASHM) On the commutativity degree of compact groups. (English summary) Arch. Math. (Basel) 93 (2009), no. 4, 345–356.

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!

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You may be interested in this paper, which answers a generalization of your question about the infinite case: http://arxiv.org/abs/1102.4353

Gene and I do not yet know how our work fits in with the larger picture.

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Dear Eduardo,

I'm not entirely sure what your question is, but I'll take it as an excuse to point out some references on the degree of commutativity of a finite group $G$. I take a (very) lay interest in this because I often set to undergraduates the problem of proving that if $G$ is nonabelian then $d(G) \leq 5/8$.

A very comprehensive discussion of this is may be found in this 1979 paper of Rusin:

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102785075

There's also a 1983 article in Eureka (the magazine of the Cambridge student maths society) by Nigel Boston called "Nearly abelian groups", which (from memory) gives a fun exposition of the same thing. I've referred to this article before, so perhaps I'll take the opportunity to wander over to the library and scan it in, since this publication is not widely available outside Cambridge.

Secondly I'd like to draw attention to a paper of Peter Neumann called "Two combinatorial problems in group theory".

MR1005821 (90f:20036) Neumann, Peter M.(4-OXQ) Two combinatorial problems in group theory. Bull. London Math. Soc. 21 (1989), no. 5, 456–458.

I chanced across it quite by accident about 10 years ago. It proves the following very nice result: if $d(G) \geq \alpha$ then there are normal subgroups $K \leq H \lhd G$ with $[G : H] \leq C_1(\alpha)$, $|K| \leq C_2(\alpha)$, and $H/K$ abelian. Roughly, the only way you can have a positive proportion of elements commuting is if $G$ is virtually (small-by-abelian).

To answer your last question, I think the following paper may be relevant.

MR1764885 (2001i:20059) Lévai, L.; Pyber, L.(H-AOS) Profinite groups with many commuting pairs or involutions. (English summary) Arch. Math. (Basel) 75 (2000), no. 1, 1–7

Update: I visited the library and scanned the Eureka article. A PDF is available here: http://www.dpmms.cam.ac.uk/~bjg23/papers/boston.pdf

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Hi Ben,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 $G= direct produc of the groups G_i$ 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 Thanks for the references and the time they devoted to this issue. –  Juan Valdez Mar 10 '11 at 14:14
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