Counting Nilpotent Pairs in Finite Groups

Let \(G\) be a finite group and let \(\nu_i(G)\) denote the proportion of ordered pairs of \(G\) that generate a subgroup of nilpotency class \(i\). Various properties of the \(\nu_i(G)\)’s are established. In particular, it is shown that \(\nu_i(G) = k_i |G|/|G|^2\) for some non-negative integer \(k_i\) and that \(\sum_{i=1}^{\infty}\nu_i\) is either \(1\) or at most \(\frac{1}{2}\) for solvable groups.