  A Hausdorff topological group G is minimal if every continuous isomorphism f
: G --> H between G and a Hausdorff topological group H is open. Clearly, every
compact Hausdorff group is minimal. It is well known that every infinite
compact Hausdorff group contains a non-trivial convergent sequence. We extend
this result to minimal abelian groups by proving that every infinite minimal
abelian group contains a non-trivial convergent sequence. Furthermore, we show
that "abelian" is essential and cannot be dropped. Indeed, for every
uncountable regular cardinal kappa we construct a Hausdorff group topology
T_kappa on the free group F(kappa) with kappa many generators having the
following properties:
  (i) (F(kappa), T_kappa) is a minimal group;
  (ii) every subset of F(kappa) of size less than kappa is T_kappa-discrete
(and thus also T_kappa-closed);
  (iii) there are no non-trivial proper T_kappa-closed normal subgroups of
F(kappa).
  In particular, all compact subsets of (F(kappa), T_kappa) are finite, and
every Hausdorff quotient group of (F(kappa), T_kappa) is minimal (that is,
(F(kappa), T_kappa) is totally minimal).
