can appear in the isolation annulus ($\theta_{G} < \theta < 3\theta_{G}$) and thus can cause bonafide systems to fail the isolation criterion.

Because we have a complete catalog in redshift space, we can apply a cleaner isolation criterion in three dimensions. We compute the isolation ratio, $\Upsilon \equiv \theta_{N}/\theta_{G}$, for all groups in the sample. $\theta_{G}$ is again the radius of the smallest circle which contains the centers of all the group members and $\theta_{N}$ is the angular radius of the largest concentric circle which contains no further galaxies within 1200 km s$^{-1}$ of the median group velocities. Our velocity separation cutoff is somewhat arbitrary; it is more relaxed than the group selection cutoff to allow for a larger velocity dispersion in the group environment. Our isolation criterion is not completely equivalent to Hickson’s criterion because we only search for surrounding galaxies brighter than the magnitude limit, $m_{\rm Zw}=15.5$. Thus, for groups with $m_{1} > 12.5$, we do not search for surrounding galaxies throughout $[m_{1},m_{1}+3]$. Twenty-five RSCG’s fail the “isolation criterion” which requires $\Upsilon > 3$. Six of these groups have $N=3$, 19 have $N \geq 4$.

We explore the differences between “isolated” groups and groups with $\Upsilon < 3$. 