The Probability That an Operator Is Nilpotent

Choose a random linear operator on a vector space of finite cardinality N; then the probability that it is nilpotent is 1 / N . This is a linear analogue of the fact that for a random self-map of a set of cardinality N, the probability that some iterate is constant is 1 / N . The first result is due to Fine, Herstein, and Hall, and the second is essentially Cayley's tree formula. We give a new proof of the result on nilpotents, analogous to Joyal's beautiful proof of Cayley's formula. It uses only general linear algebra and avoids calculation entirely.