In 1984, Cohen and Lenstra proposed a "random group" model
for ideal class groups of quadratic imaginary extensions of the
rational numbers. They computed many statistics of their
model---notably the precise probability that a finite abelian group
has size divisible by three. This probability is (perhaps
surprisingly) greater than one-third! (Analogous surprises hold for
other primes.) In joint work with John Duncan, we compute the
probability that a finite abelian group has size congruent one mod
three (and other analogous results).