## Commuting elements of non-abelian groups

The following is simply a collection of results, which I haven’t checked  just yet.

Let $G$ be a finite non-abelian group.  Let $N_c(G)$ be the number of ordered pairs $(x,y)\in G\times G$ such that $xy=yx$,   and $N_t(G)$  be the total number of ordered pairs in $G\times G$.  So the real number $Pr(G):=N_c(G)/N_t(G)$ belongs to $[0,1]$.

In 1979, Rusin has worked out explicitely the groups for which $Pr(G)>11/32$, in this freely available paper.  The maximum value is $5/8$, attained both by the quaternion group $H_8$ and the dihedral group $D_4$.   Then there is an infinite decreasing series starting at $5/8$ and accumulating at $1/2$.  And finally between $1/2$ and $11/32$ there are a handful of other cases.

Some problems regarding lower bounds, or the density of the values of $Pr(G)$ in intervals $[a,b]$ with $b\leq 1/4$ are mentionned, I believe they are still open.

The problem has been generalized more recently:   to n-tuples in compact infinite groups by Erfanian and Russo,   to pairs of  subgroups by Tărnăuceanu,  and to semigroups by Givens (in which case there is density in $[0,1]$.)