  In the paper we introduce the construction of invariants for 3-manifolds,
based on the same key concepts as the classical Dijkgraaf-Witten invariant. We
introduce the notion of a special $G$-system and describe how each system
induces the invariant not only for closed 3-manifolds, but also for manifolds
with boundary. Finally, we show how to construct a very simple one-dimensional
special $G$-system using group cohomologies.
