  For an arbitrary group $G$, it is shown that either the semigroup rank $G{\rm
rk}S$ equals the group rank $G{\rm rk}G$, or $G{\rm rk}S = G{\rm rk}G+1$. This
is the starting point for the rest of the article, where the semigroup rank for
diverse kinds of groups is analysed. The semigroup rank of relatively free
groups, for any variety of groups, is computed. For a finitely generated
abelian group~$G$, it is proven that $G{\rm rk}S = G{\rm rk}G+1$ if and only if
$G$ is torsion-free. In general, this is not true. Partial results are obtained
in the nilpotent case. It is also proven that if $M$ is a connected closed
surface, then $(\pi_1(M)){\rm rk}S = (\pi_1(M)){\rm rk}G+1$ if and only if $M$
is orientable.
