The distribution of group structures on elliptic curves over finite prime fields

We determine the probability that a randomly chosen elliptic curve E/F-p over a randomly chosen prime field F-p has an l-primary part E (F-p) [l infinity] isomorphic with a fixed abelian l-group H ((l)) (alpha,beta) = Z/ l(alpha) x Z/l(beta). Probabilities for "|E(F-p)| divisible by n", "E(F-p) cyclic" and expectations for the number of elements of precise order n in E(Fp) are derived, both for unbiased E/F-p and for E/F-p with p equivalent to 1 (l(r)).