Relativistic electron fluxes observed in geosynchronous orbit by GOES-8 in 1997 to 2000 were considered as a complex function of geomagnetic indices PC, Kp, and Dst, as well as parameters of the magnetosphere size, subsolar Rs, and terminator Rf magnetopause distances. A geosynchronous relativistic electron enhancement (GREE) is determined as daily maximal electron flux exceeding the upper root mean square deviation (RMSD) threshold of about 1500 (cm2s sr)-1. Comparison analysis of the GREE . dynamics and geomagnetic conditions on the rising phase of current solar cycle revealed suppression of the relativistic electron enhancements by substantially increased strong geomagnetic activity in the solar maximum. Statistical consideration of a relationship between the GREEs and the geomagnetic parameters showed that the most important parameters controlling the geosynchronous relativistic electron enhancements were 4-day averaged Kp index, PC index, and magnetopause termination distance Rf, delayed on 3 and 14 hours, respectively. Relatively high averaging time for Kp was explained by the cumulative effect of substorm energy release in a gradual mechanism accelerating the relativistic electrons in the magnetosphere. Very short time delay for PC index was interpreted as intensification of a fast acceleration mechanism producing the GREEs during severe geomagnetic storms. Substantial increase of the PC index (PC > 5) was found to be a sufficient condition for GREE occurrence. The fast response of the geosynchronous relativistic electron fluxes on the magnetosphere compression was explained by drift losses of the energetic electrons at the magnetopause, which approaches the Earth during geomagnetic storms.