  In this paper, we study gravitational lensing by groups of galaxies. Since
groups are abundant and therefore have a large covering fraction on the sky,
lensing by groups is likely to be very important observationally. Besides, it
has recently become clear that many lens models for strong lensing by
individual galaxies require external shear to reproduce the observed image
geometries; in many cases a nearby group is detected that could provide this
shear. In this work, we study the expected lensing behavior of galaxy groups in
both the weak and strong lensing regime. We examine the shear and magnification
produced by a group and its dependence on the detailed mass distribution within
the group. We find that the peak value of the weak lensing shear signal is of
the order of 3 per cent and varies by a factor of about 2 for different mass
distributions. These variations are large enough to be detectable in the Sloan
Digital Sky Survey (SDSS). In the strong lensing regime we find that the image
geometries and typical magnifications are sensitive to the group properties and
that groups can easily provide enough external shear to produce quadruple
images. We investigate the statistical properties of lensing galaxies that are
near or part of a group and find that statistical lens properties, like the
distribution of time delays, are affected measurably by the presence of the
group which can therefore introduce an additional systematic error in the
measurement of the Hubble constant from such systems. We conclude that both the
detection of weak lensing by groups and accurate observations of strong galaxy
lens systems near groups could provide important information on the total mass
and matter distribution within galaxy groups.
