We examine the scalings of X-ray luminosity, temperature, and dark matter or galaxy velocity dispersion for galaxy groups in a ΛCDM cosmological simulation, which incorporates gravity, gas dynamics, radiative cooling, and star formation, but no substantial nongravitational heating. In agreement with observations, the simulated LX-σ and LX-TX relations are steeper than those predicted by adiabatic simulations or self-similar models, with LX σ4.4 and LX T for massive groups and significantly steeper relations below a break at σ 180 km s-1 (TX 0.7 keV). The TX-σ relation is fairly close to the self-similar scaling relation, with TX σ1.75, provided that the velocity dispersion is estimated from the dark matter or from 10 galaxies. The entropy of hot gas in low-mass groups is higher than predicted by self-similar scaling or adiabatic simulations, and it agrees with observational data that suggest an "entropy floor." The steeper scalings of the luminosity relations are driven by radiative cooling, which reduces the hot (X-ray-emitting) gas fraction from 50% of the total baryons at σ 500 km s-1 to 20% at σ 100 km s-1. A secondary effect is that hot gas in smaller systems is less clumpy, further driving down LX. A smaller volume simulation with 8 times higher mass resolution predicts nearly identical X-ray luminosities at a given group mass, demonstrating the insensitivity of the predicted scaling relations to numerical resolution. The higher resolution simulation predicts higher hot gas fractions at a given group mass, and these predicted fractions are in excellent agreement with available observations. There remain some quantitative discrepancies: the predicted mass scale of the LX-TX and LX-σ breaks is somewhat too low, and the luminosity-weighted temperatures are too high at a given σ, probably because our simulated temperature profiles are flat or rising toward small radii, while observed profiles decline at r 0.2Rvir. We conclude that radiative cooling has an important quantitative impact on group X-ray properties and can account for many of the observed trends that have been interpreted as evidence for nongravitational heating. Improved simulations and observations are needed to understand the remaining discrepancies and to decide the relative importance of cooling and nongravitational heating in determining X-ray scalings.