  We investigate the use of the cross-correlation between galaxies and galaxy
groups to measure redshift-space distortions (RSD) and thus probe the growth
rate of cosmological structure. This is compared to the classical approach
based on using galaxy auto-correlation. We make use of realistic simulated
galaxy catalogues that have been constructed by populating simulated dark
matter haloes with galaxies through halo occupation prescriptions. We adapt the
classical RSD dispersion model to the case of the group-galaxy
cross-correlation function and estimate the RSD parameter $\beta$ by fitting
both the full anisotropic correlation function $\xi(r_p,\pi)$ and its multipole
moments. In addition, we define a modified version of the latter statistics by
truncating the multipole moments to exclude strongly non-linear distortions at
small transverse scales. We fit these three observable quantities in our set of
simulated galaxy catalogues and estimate statistical and systematic errors on
$\beta$ for the case of galaxy-galaxy, group-group, and group-galaxy
correlation functions. When ignoring off-diagonal elements of the covariance
matrix in the fitting, the truncated multipole moments of the group-galaxy
cross-correlation function provide the most accurate estimate, with systematic
errors below 3% when fitting transverse scales larger than $10Mpc/h$. Including
the full data covariance enlarges statistical errors but keep unchanged the
level of systematic error. Although statistical errors are generally larger for
groups, the use of group-galaxy cross-correlation can potentially allow the
reduction of systematics while using simple linear or dispersion models.
