When is a semigroup a group?

A well-known necessary and sufficient condition for the operator A to be the infinitesimal generator of a strongly continuous (C
0
) group is that both A and −A generate a C
0
-semigroup. This seems to imply that one has to check the conditions in the Hille-Yosida Theorem for both A and −A. In this paper we show that this is not necessary. Given that A generates a C
0
-semigroup we prove that a (weak) growth bound on the resolvent on a left half plane is sufficient to guarantee that A generates a group. This extends the recent result found by Liu, see [Liu98].