The Group Evolution Multiwavelength Study (GEMS) involves a multiwavelength study of a sample of 60 galaxy groups, chosen to span a wide range of group properties. Substantial ROSAT Position Sensitive Proportional Counter (PSPC) observations, available for all of these groups, are used to characterize the state of the intergalactic medium in each. We present the results of a uniform analysis of these ROSAT data and a statistical investigation of the relationship between X-ray and optical properties across the sample. Our analysis improves in several respects on previous work: (i) we distinguish between systems in which the hot gas is a group-scale medium and those in which it appears to be just a hot halo associated with a central galaxy; (ii) we extrapolate X-ray luminosities to a fixed overdensity radius (r(500)) using fitted surface brightness models, in order to avoid biases arising from the fact that cooler systems are detectable to smaller radii, and (iii) optical properties have been rederived in a uniform manner from the NASA Extragalactic Database, rather than relying on the data in the disparate collection of group catalogues from which our systems are drawn. The steepening of the L-X-T-X relation in the group regime reported previously is not seen in our sample, which fits well on to the cluster trend, albeit with large non-statistical scatter. A number of biases affect the fitting of regression lines under these circumstances, and until the impact of these has been thoroughly investigated it seems best to regard the slope of the Group L-X-T-X relation as being poorly determined. A significant problem in comparing the properties of groups and clusters is the derivation of system radii, to allow different systems to be compared within regions having the same overdensity. We find evidence that group velocity dispersion (sigma(v)) provides a very unreliable measure of system mass (and hence radius), with a number of groups having remarkably low values of sigma(v), given that they appear from their X-ray properties to be collapsed systems. We confirm that the surface brightness profiles of groups are significantly flatter than those of clusters - the maximum value of the beta(fit) parameter for our sample is 0.58, lower than the typical value of 0.67 seen in clusters - however, we find no significant tendency within our sample for cooler groups to show flatter profiles. This result is inconsistent with simple universal pre-heating models. The morphology of the galaxies in the GEMS groups is correlated to their X-ray properties in a number of ways: we confirm the very strong relationship between X-ray emission and a dominant early-type central galaxy, which has been noted since the early X-ray studies of groups, and also find that spiral fraction is correlated with the temperature of the hot gas and hence the depth of the gravitational potential. A class of spiral-rich groups with little or no X-ray emission probably corresponds to groups that have not yet fully collapsed.