We study the evolution of galaxies inside and outside of the group environment since z=1 using a large well defined set of groups and galaxies from the zCOSMOS-bright redshift survey in the COSMOS field. The fraction of galaxies with early-type morphologies increases monotonically with M_B luminosity and stellar mass and with cosmic epoch. It is higher in the groups than elsewhere, especially at later epochs. The emerging environmental effect is superposed on a strong global mass-driven evolution, and at z~0.5 and log(M*/Msol)~10.2, the effect of group environment is equivalent to (only) about 0.2 dex in stellar mass or 2 Gyr in time. The stellar mass function of galaxies in groups is enriched in massive galaxies. We directly determine the transformation rates from late to early morphologies, and for transformations involving colour and star formation indicators. The transformation rates are systematically about twice as high in the groups as outside, or up to 3-4 times higher correcting for infall and the appearance of new groups. The rates reach values, for masses around the crossing mass 10^10.5 Msol, as high as (0.3-0.7)/Gyr in the groups, implying transformation timescales of 1.4-3 Gyr, compared with less than 0.2/Gyr, i.e. timescales >5 Gyr, outside of groups. All three transformation rates decrease at higher stellar masses, and must decrease also at the lower masses below 10^10 Msol which we cannot well probe. The rates involving colour and star formation are consistently higher than those for morphology, by a factor of about 50%. Our conclusion is that the transformations which drive the evolution of the overall galaxy population since z~1 must occur at a rate 2-4 times higher in groups than outside of them.
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