It has been known that an antiunitary symmetry such as time-reversal or charge conjugation is needed to realize Z(2) topological phases in noninteracting systems. Topological insulators and superconducting nanowires are representative examples of such Z(2) topological matters. Here we report the Z(2) topological phase protected by only unitary symmetries. We show that the presence of a nonsymmorphic space group symmetry opens a possibility to realize Z(2) topological phases without assuming any antiunitary symmetry. The Z(2) topological phases are constructed in various dimensions, which are closely related to each other by Hamiltonian mapping. In two and three dimensions, the Z(2) phases have a surface consistent with the nonsymmorphic space group symmetry, and thus they support topological gapless surface states. Remarkably, the surface states have a unique energy dispersion with the Mobius twist, which identifies the Z(2) phases experimentally. We also provide the relevant structure in the K theory.