  A ring $R$ with center $C$ is said to be \textit{centrally essential} if the
module $R_C$ is an essential extension of the module $C_C$. In the paper, we
study groups whose group algebras over fields are centrally essential rings. We
focus on the centrally essential modular group algebras of finite groups over
fields of nonzero characteristic. V.T.Markov is supported by the Russian
Foundation for Basic Research, project 17-01-00895-A. A.A.Tuganbaev is
supported by Russian Scientific Foundation, project 16-11-10013.
