A nilpotent group G is fgp if Gp, is finitely generated (fg) as a p-local group for all primes
p; it is fg-like if there exists a nilpotent fg group H such that Gp≃Hp for all primes p. The fgp nilpotent
groups form a (generalized) Serre class; the fg-like nilpotent groups do not. However, for abelian
groups, a subgroup of an fg-like group is fg-like, and an extension of an fg-like group by an fg-like group
is fg-like. These properties persist for nilpotent groups with finite commutator subgroup, but fail in
general.