Let G be a locally compact group and let \((\mu_ t)_{t\geq 0}\) be a continuous semigroup of bounded measures with \(\| \mu_ t\| \leq 1\), \(\mu_ 0=\epsilon_ 0\). Let \((A_ t)\) be the corresponding semigroup of left invariant contractions \(f\to \mu_ t*f\) on \(C_ 0(G)\). The generator \(T_ A\) of \((A_ t)\) is left invariant, dissipative, closed and densely defined. Conversely, it is known that every left invariant, dissipative, densely defined operator on \(C_ 0(G)\) admits an extension to a generator of a continuous left invariant contraction semigroup \((A_ t)\). [Cf. \textit{J.-P. Roth}, Ann. Inst. Fourier 26 (1976), No.4, 1-97 (1977; Zbl 0331.47021).] Now the problem arises, if every closed densely defined dissipative and left invariant operator T is already generator of a semigroup \((A_ t).\) The answer is affirmative if the domain D(T) contains a dense left and right invariant subspace [see e.g. \textit{F. Hirsch}, Sémin. Théorie Potent., 15e année 1972, Exp. 16 (1973; Zbl 0322.31013); \textit{J. Faraut} and \textit{K. Harzallah}, Ann. Inst. Fourier 22, No.2, 147-164 (1972; Zbl 0226.46027); the reviewer, Stetige Faltungshalbgruppen von Wahrscheinlichkeitsmaßen und erzeugende Distributionen (Lect. Notes Math. 595) (1977; Zbl 0373.60010)], hence especially for compact and for abelian groups. The author shows that the answer also is affirmative if G contains an open subgroup which is a direct product of an abelian and a compact group, but in general the answer is negative. This is shown by an example which is constructed on the 3-dimensional Heisenberg group \({\mathbb{H}}_ 1\).