  Given a group G, we construct, in a canonical way, an inverse semigroup S(G)
associated to G. The actions of S(G) are shown to be in one-to-one
correspondence with the partial actions of G, both in the case of actions on a
set, and that of actions as operators on a Hilbert space. In other words, G and
S(G) have the same representation theory.
  We show that S(G) governs the subsemigroup of all closed linear subspaces of
a G-graded C*-algebra, generated by the grading subspaces. In the special case
of finite groups, the maximum number of such subspaces is computed.
  A ``partial'' version of the group C*-algebra of a discrete group is
introduced. While the usual group C*-algebra of finite commutative groups
forgets everything but the order of the group, we show that the partial group
C*-algebra of the two commutative groups of order four, namely Z/4 and Z/2+Z/2,
are not isomorphic.
