  Many catalogues of isolated compact groups of galaxies (CGs) have been
extracted using Hickson's criteria to identify isolated, dense systems of
galaxies, with at least three or four galaxies concordant in magnitude and
redshift. But is not clear to what extent the catalogues of CGs are complete
and reliable, relative to 3D truly isolated, dense groups. Using five different
semi-analytical models of galaxy formation (SAMs), we identify isolated dense
groups in 3D real space, containing at least three galaxies. We then build mock
redshift space galaxy catalogues and run a Hickson-like CG finder. We find that
the Hickson-like algorithm in redshift space is poor at recovering 3D CGs of at
least 3 galaxies, with a purity of $\sim 10\%$ and a completeness of $\sim
22\%$. Among the $\sim 90\%$ of spurious systems, typically $60\%$ are dense
structures that failed the 3D isolation criteria, while the remaining $40\%$
are chance alignments of galaxies along the line of sight, nearly all of which
are within regular groups, with some variation with the SAM used for the
analysis. In other words, while only $10\%$ of CGs are isolated dense groups,
as intended, half are dense structures embedded within larger groups, and
one-third are chance alignments within larger groups. The low completeness of
the extracted CG sample is mainly due to the flux limits of the selection
criteria. Our results suggest that a new observational algorithm to identify
compact groups in redshift space is required to obtain dense isolated galaxy
systems.
