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].)


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: