In 1964 Weeg [6] has posed a question asking which semi-groups admit strongly connected automata. The presumed answer was given in 1971 by Oehmke [5] through the following statement: A semigroup admits strongly connected automaton if it has a homomorphic image either a nontrivial finite group or semigroup of the transformations of some right zero semigroup. This theorem can be reformulated as follows: A semigroup is a characteristic semigroup of strongly connected automaton if it has a homomorphic image either a nontrivial finite group or a semigroup of transformations of some right zero semigroup. In this paper the "only if" part of this theorem has been overthrown. Some related problems concerning strongly connectedness of finite automata are also investigated. Key Words and Phrases: semigroup, characteristic semigroup of an automaton, strongly connected automaton.