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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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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