Two characterizations of finite nilpotent groups

Abstract

Abstract In this note we give two characterizations of finite nilpotent groups. First, we show that a finite group G is not p-nilpotent if and only if it contains two elements of order q k {q^{k}} , for q a prime different than p, whose product has order p or possibly 4 if p = 2 {p=2} . We also show that the set of words on two variables where the total degree of each variable is ± 1 {\pm 1} can be used to characterize finite nilpotent groups. Using this characterization we show that if a finite group is not nilpotent, then there is a word map of specified form for which the corresponding probability distribution is not uniform.