*[Corrected, thanks – T.]*

For more details see

http://www.renyi.hu/~endre/growth.pdf ]]>

The article is by Nigel Boston and it is in volume 43 of Eureka, published in Cambridge in 1983. The title is “nearly abelian groups”. It may well be an exposition of the paper of Rusin you mention above.

Ben

]]>This maximum of 5/8 is attained for both the quaternion group and dihedral group , two groups of order 8.

The general issue of evaluating the probability that two elements of a nonabelian finite group commute is discussed in great detail in a 1979 paper in *Pacific J. Math.* by Rusin which is freely available here. (Unfortunately, I couldn’t find a reference online for the “Eureka” magazine you mentionned.)

It appears that there’s indeed an infinite series starting at 5/8 and accumulating at 1/2. The paper mentions several open problems regarding the possible density of in intervals [a,b] with , which seem still open (at least a quick glance at google scholar suggests so).

]]>I believe that in a non-abelian group at most 5/8 of the pairs of elements can commute. This is a fun exercise in undergraduate group theory.

After that there is a jump to the next smallest possible proportion, and then a discrete series which accumulates somewhere; maybe 1/2, though I forget.

There is a fun article on this in an old edition (circa 1970) of “Eureka”, the magazine of the Cambridge Student maths society. As I’m in Princeton right now, I can’t supply the reference – maybe someone else can?

Ben

]]>