Two characterizations of finite nilpotent groups

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), for q a prime different than p, whose product has order p or possibly 4 if p = 2. We also show that the set of words on two variables where the total degree of each variable is +/- 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.