  We give three necessary and sufficient conditions for a pro-p group to be
p-adic analytic. We show that a noetherian pro-p group having finite chain
length has a finite rank and conversely. We further deduce that a noetherian
pro-p group has a finite rank precisely when it satisfies the weak descending
chain condition. Using these results, we resolve a conjecture posed by Lubotzky
and Mann in the affirmative within the class of noetherian groups which are
countably based.
  Using these results, we answer a related conjecture about pro-p groups for
the case of countably based pro-p groups. Namely, we prove that if every closed
but non-open subgroup of a countably based pro-p group has finite rank, then
the group is p-adic analytic and conversely.
