Isoclinism classes and commutativity degrees of finite groups
Let \(G\) be a finite group, and for \(n\geq 0\), define \[ d_n(G)=|G|^{-(n+1)}|\{(x_1,\dots,x_{n+1})\in G^{n+1}\mid[x_i,x_j]=1,\;1\leq i,j\leq n+1\}|. \] So, \(d_1(G)=d(G)\) is the probability that two elements chosen randomly from \(G\) (with replacement) commute [see the reviewer, Am. Math. Mon. 80, 1031-1034 (1973; Zbl 0276.60013)]. It is shown that \(d_n(G)=d_n(H)\) if \(G\) and \(H\) are isoclinic in the sense of \textit{P. Hall} [J. Reine Angew. Math. 182, 130-141 (1940; Zbl 0023.21001)]. It is then shown that if \(G\) is nonabelian, \(d_n(G)\leq(3\cdot 2^n-1)/2^{2n+1}\), with equality if and only if \(G\) is isoclinic to the quaternion group of order eight. A number of old and new results about \(d(G)\) are also presented. In particular, the groups with \(d(G)\geq 1/2\) are characterised, as are the non-nilpotent groups with \(d(G)=1/2\). Some of the author's results, including the bound on \(d_n(G)\), have previously been published in a journal that is not widely available.
scientific article; zbMATH DE number 827379