In the field of symmetry-protected topological phases, a common wisdom is that the symmetries may fix the nontrivial topological classifications, but they alone cannot determine whether a system is topologically nontrivial. Here, we show that this is no longer true in cases where symmetries are projectively represented. Particularly, the Zak phase, a topological invariant of a one-dimensional system, can be entirely determined by the projective symmetry algebra (PSA). To demonstrate this remarkable effect, we propose a minimal model, termed the flux Su-Schrieffer-Heeger (f-SSH) model, where the bond dimerization in the original SSH model is replaced by a flux dimerization. We present experimental realization of the f-SSH model in an electric-circuit array, and our predictions are directly confirmed by electric measurement. Our work refreshes the understanding of the relationship between symmetry and topology, opens up avenues for exploring novel PSA determined topological phases, and suggests flux dimerization as an approach for designing topological crystals.