Summary: Two binary operations on the Lorentzian sphere in \(\mathbb{R}^4_1\) and on the Lorentzian cylinder in \(\mathbb{R}^5_1\) are denned. Also, it is shown that the sphere in \(\mathbb{R}^4_1\) and the Lorentzian cylinder in \(\mathbb{R}^5_1\) with the corresponding binary operations form Lie groups. A \(C^\infty\)-action on the Lorentzian cylinder of arbitrary radius and of Lorentzian cylinder of radius one is defined in \(\mathbb{R}^5_1\) and some properties are given. Finally, the action is expanded to \(\mathbb{R}^5_1\).