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.
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