Counting Nilpotent Pairs in Finite Groups: Some Conjectures

The number of nilpotent pairs is determined for a number of small groups

Counting Nilpotent Pairs in Finite Groups

Let G be a finite group and let i (G) denote the proportion of ordered pairs of G that generate a subgroup of nilpotency class i. Various properties of the i 's are established. In particular it is shown that i = k i \Delta jGj=jGj 2 for some non-negative integers k i and that P 1 i=1 i is either 1 or at most 1/2 for solvable groups. 1 Introduction Let G be a finite group and let i (G) = n i (G) jGj 2 where n i (G) = jf(x; y) 2 G 2 jhx; yi is nilpotent of class igj for 0 i 1. We take `hx; yi is nilpotent of class 0' to mean that hx; yi is non-nilpotent. Clearly, 0 (G) = 1 \Gamma 1 X i=1 i (G): It is well known that 1 (G), the proportion of commuting pairs in G, is at most 5/8 for non-abelian groups [5]. There is no analogous lower bound for 1 (G). In particular, 1 (S n ) ! 0 where S n is the All authors' work supported by NSF grant NSF-DMS 9100509 symmetric group on n symbols. Both of these results follow from the fact that 1 (G) is the ratio of th..