P-nilpotent completion is not idempotent.
Let P be an arbitrary set of primes. The P-nilpotent completion of a group G is defined by the group homomorphism η: G → G where G = inv lim(G/ΓG). Here ΓG is the commutator subgroup [G,G] and ΓG the subgroup [G, ΓG] when i > 2. In this paper, we prove that P-nilpotent completion of an infinitely generated free group F does not induce an isomorphism on the first homology group with Z coefficients. Hence, P-nilpotent completion is not idempotent. Another important consequence of the result...