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This is with regard 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!
show/hide this revision's text 1

This is with regard 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!