  A group is small if it has countably many complete $n$-types over the empty
set for each natural number n. More generally, a group $G$ is weakly small if
it has countably many complete 1-types over every finite subset of G. We show
here that in a weakly small group, subgroups which are definable with
parameters lying in a finitely generated algebraic closure satisfy the
descending chain conditions for their traces in any finitely generated
algebraic closure. An infinite weakly small group has an infinite abelian
subgroup, which may not be definable. A small nilpotent group is the central
product of a definable divisible group with a definable one of bounded
exponent. In a group with simple theory, any set of pairwise commuting elements
is contained in a definable finite-by-abelian subgroup. First corollary : a
weakly small group with simple theory has an infinite definable
finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal
solvable group A of derived length n is contained in an A-definable almost
solvable group of class n.
