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This paper investigates the control role of the relative phase between the probe and driving fields on the gain,dispersion and populations in an open Λ system with spontaneously generated coherence (SGC) .It shows that by adjusting the value of the relative phase,a change from lasing with inversion to lasing without inversion can be realized;the values and frequency spectrum regions of the inversionless gain and dispersion can be obviously varied;high refractive index with zero absorption and electromagnetically induced transparency can be achieved.It is also found that when the driving field is resonant,the shapes of the dispersion and the gain curves versus the probe detuning are very similar if the relative phase of the dispersion lags π/2 than that of the gain,however for the off-resonant driving field the similarity will disappear;the gain,dispersion and populations are periodical functions of the relative phase,the modulation period is always 2π:the contribution of SGC to the inversionless gain and dispersion is much larger than that of the dynamically induced coherence.