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We identify two generalizations of the notion of a finitely generated nilpotent. Thus a nilpotent group G is fgp if Gp is fg as p-local group for each p; and G is fg-like if there exists a fg nilpotent group H such that Gp ≅ Hp for all p. The we have proper set-inclusions:
{fg} ⊂ {fg-like} ⊂ {fgp}.
We examine the extent to which fg-like nilpotent groups satisfy the axioms for a Serre class....
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